Patent application title: System and method for using acoustic field parameters for communication

Abstract:

A system and a method for conveying data using the vector components of
the acoustic field, in addition to the scalar component is described.
Receivers capable of determining the transmitted data values by measuring
one or more parameters of an acoustic field are utilized. Measured
parameters of the acoustic field may include both the scalar and vector
components. The described system and method may optimize the conveyance
of communications through a fluid.

Claims:

1. A method, comprising:providing a communication system having at least
one pressure transducer and operating in a fluid medium;transmitting data
through the fluid medium using the at least one pressure transducer;
andreceiving the data using at least one vector sensor.

2. The method of claim 1 wherein the fluid medium includes water.

3. The method of claim 1 further comprising:the vector sensor receiving
the data on a plurality of respective channels thereof.

4. The method of claim 1 further comprising the vector sensor receiving
the data on one scalar pressure channel, and a plurality of velocity
measurement channels.

5. The method of claim 1 further comprising:using a plurality of pressure
transducers to transmit the data; andusing only one vector sensor to
receive the data.

6. The method of claim 1 further comprising:using a given number of
pressure transducers to transmit the data; andusing a number of vector
sensors, that is less than the given number of pressure transducers, to
receive the data.

8. The method of claim 4 further comprising:the vector sensor measuring a
plurality of orthogonal particle velocity components using a respective
plurality of channels.

9. The method of claim 4 further comprising:the vector sensor measuring
particle velocity components along the X, Y, and Z directions at three
respective channels of the vector sensor.

10. The method of claim 1 wherein the at least one vector sensor is
operable to measure thirteen separate measurements on thirteen respective
channels.

11. The method of claim 10 wherein the thirteen measurements include: one
scalar pressure measurement; three particle velocity measurements along
three respective directions; and the variations of velocity as a function
of position along three axes, for each of the three velocity directions.

12. A communication system comprising:at least one pressure transducer
operating in a fluid medium and operable to transmit data through the
medium; andat least one vector sensor operable to receive the data.

13. The system of claim 12 wherein the fluid medium includes water.

14. The method of claim 12 wherein the vector sensor is operable to
receive the data on a plurality of respective channels thereof.

15. The system of claim 12 wherein the vector sensor is operable to
receive data on one scalar pressure channel, and a plurality of velocity
measurement channels.

16. The system of claim 12 wherein the at least one pressure transducer
comprises:a plurality of pressure transducers; andthe at least one vector
sensor comprises only one vector sensor.

17. The system of claim 12 wherein the at least one pressure transducer
comprises a given number of pressure transducers; andthe at least one
vector sensor comprises a number of vector sensors that is smaller than
the given number of pressure transducers.

18. The system of claim 16 wherein the plurality of pressure transducers
transmit their respective data simultaneously.

19. The system of claim 15 wherein the vector sensor is operable to
measure a plurality of orthogonal particle velocity components using a
respective plurality of channels.

20. The system of claim 16 wherein the vector sensor is operable to
measure particle velocity components along the X, Y, and Z directions at
three respective channels of the vector sensor.

21. The system of claim 12 wherein the at least one vector sensor is
operable to measure thirteen separate measurements on thirteen respective
channels thereof.

22. The system of claim 21 wherein the thirteen measurements include: one
scalar pressure measurement; three particle velocity measurements along
three respective directions; and the variations of velocity as a function
of position along three axes, for each of the three velocity directions.

Description:

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001]This application claims the benefit of U.S. Provisional Patent
Application Ser. No. 60/835,408, filed Aug. 3, 2006, entitled "System and
Method for Using Acoustic Field Parameters in Communication" the entire
disclosure of which is hereby incorporated by reference herein.

BACKGROUND OF THE INVENTION

[0002]The present invention relates to the field of acoustic
communications. More particularly, the invention relates to the
measurement of one or more parameters of the acoustic field for
information recovery.

[0003]Acoustic waves have been used for target localization and SONAR
applications for many years, especially in underwater applications. The
steady growth of ocean exploration activity in recent years has resulted
in a rising need to convey data through underwater channels. Numerous
applications of acoustic communications pose an increasing demand on
high-speed underwater wireless telemetry and data communication systems.
These systems often require a combination of sensors, autonomous
underwater vehicles, moored instruments, and/or surface ships to
communicate with each other. Examples of such applications include:
real-time remote monitoring of underwater tools, construction, and/or
environmental factors in the offshore oil industry, continuous
observation of ocean phenomena over geographically large areas,
observation of fisheries, as well as many naval and security
applications, including, but not limited to harbor monitoring systems and
tactical surveillance operations.

[0004]Underwater communication systems generally use acoustic waves to
convey information. In the underwater environment, electromagnetic waves
do not propagate, as they attenuate rapidly.

[0005]In general, underwater acoustic channels are bandwidth-constrained.
For distances from 10 km up to 100 km (long range), the available
bandwidth is about a few kHz, whereas in a 1-10 km medium range setup,
the available bandwidth is almost a few 10 kHz. Communications over short
ranges, smaller than around 100 m, may have an available bandwidth
exceeding 100 kHz. Underwater communication may be complicated by the
harsh multipath conditions, and/or channel-alone time variations due to
water surface fluctuations, internal waves, and/or turbulence. The
multipath conditions may result in delay spreads up to several hundreds
of symbols for high data rates. Further, channel-alone time variations
may result in Doppler spreads up to several 10 Hz. After the first
generation of underwater (analog) modems, second generation (digital)
modems used non-coherent techniques such as frequency shift keying (FSK)
and differentially coherent schemes like differential phase shift keying
(DPSK). Due to the need for higher spectral efficiencies over typical
channels of interest, coherent systems with phase shift keying (PSK) and
quadrature amplitude modulation (QAM) were also developed.

[0006]Data rates available from existing systems are much lower than the
data rates required for the real-time transmission of data, such as video
and telemetry signals, over medium and long distances. For example, a
typical commercially available modem provides only up to 2400 b/s at a 2
km depth and 3 km range setup.

[0007]Traditionally, underwater acoustic transmission has been limited to
the scalar component of the acoustic field, i.e., the pressure. Existing
multichannel underwater receivers are, generally, composed of spatially
separated pressure-only sensors resulting in large size arrays. Array
size is a limitation in modem applications, especially for small
autonomous underwater vehicles. For example, the medium frequency (MF) 3
kHz receive array of a modem designed for a 21-inch diameter autonomous
underwater vehicle includes four hydrophones and is 1.5 m long. For
smaller size autonomous underwater vehicles, the necessary modem array
may prove unwieldy.

[0008]In the past few decades, a large volume of research has been
conducted on theory, performance evaluation, and design of acoustic
vector sensors. These acoustic vector sensors have been used for the
detection of acoustic signals, for example, underwater target
localization and SONAR applications. For example, vector sensors have
been studied for use in applications including accurate azimuth and
elevation estimation of a source, avoidance of the left-right ambiguity
of linear towed arrays of scalar sensors, and acoustic noise reduction
due to a highly directive beam pattern.

[0009]The presently disclosed novel system and method include all of the
same advantages present in traditional techniques but eliminate
associated disadvantages.

SUMMARY OF THE INVENTION

[0010]According to one aspect, the invention provides a method, that may
include providing a communication system having at least one pressure
transducer and operating in a fluid medium;

[0011]transmitting data through the fluid medium using the at least one
pressure transducer; and receiving the data using at least one vector
sensor.

[0012]According to another aspect, the invention provides a communication
system, that may include at least one pressure transducer operating in a
fluid medium and operable to transmit the data through the fluid medium;
and at least one vector sensor operable to receive the data.

[0013]The invention describes a method for measuring scalar and/or vector
components of an acoustic field. The acoustic field may travel through
any medium. A scalar component of an acoustic field is the pressure. A
vector component of an acoustic field includes measurements of particle
motion including derivatives of the displacement of particles. This may
include a spatial derivative of pressure. The spatial derivative of the
pressure may be referred to as a pressure gradient or, in some
circumstances, the velocity or acoustic particle velocity. In the
following description these terms may be used interchangeably. In
addition, vector components used may include spatial derivatives of
velocity, velocity gradients, or any higher order gradients. Velocity
gradients may be referred to as acceleration in the following disclosure
and are to be considered equivalent.

[0014]Systems utilizing parameters of an acoustic field to identify data
may use any combination of scalar and vector components. For example, the
three orthogonal components of velocity and the scalar pressure at a
single point can be used to recover information from an acoustic field.
Receivers, such as vector sensors, can be efficiently manufactured today
and enable the use of the unexplored degrees of freedom of the channel.
Utilizing vector components of the acoustic field reduces the array size
needed to recover data transmitted, when compared with systems measuring
the scalar component alone.

[0015]Other aspects, features, advantages, etc. will become apparent to
one skilled in the art when the description of the preferred embodiments
of the invention herein is taken in conjunction with the accompanying
drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0016]For the purposes of illustrating the various aspects of the
invention, there are shown in the drawings forms that are presently
preferred, it being understood, however, that the invention is not
limited to the precise arrangements and instrumentalities shown.

[0017]FIG. 1 is a schematic representation of an acoustic communication
system, having two pressure transmitters and one vector sensor receiver;

[0019]FIG. 3 is a geometric representation of the rays received at the two
vector sensor receivers of FIG. 2, in a shallow water channel;

[0020]FIG. 4 is a 1×3 vector sensor communication system, with one
pressure transmitter and one vector sensor receiver;

[0021]FIG. 5 is graphical depiction of a performance comparison between a
single vector sensor receiver, a single pressure sensor receiver, and a
receive array with two pressure sensors, all in a frequency-flat channel;

[0022]FIG. 6 is graphical depiction of a performance comparison between a
vector sensor receiver, a single pressure sensor receiver, and a linear
receiver array with three pressure sensors, all in a frequency-selective
channel;

[0023]FIG. 7 is a geographic representation of the underwater acoustic
propagation model in the Matlab acoustic Toolbox (ACT);

[0025]FIG. 9 depicts screenshots of menus or specifying the locations of
the transmitter and the receiver, when conducting a simulation in
accordance with an embodiment of the invention;

[0026]FIG. 10 includes screenshots of a graph of a receiver impulse
response as a function of time at a given location and of a popup screen
query, in accordance with an embodiment of the invention;

[0027]FIG. 11 is a schematic illustration of an example of the time
varying discrete-time impulse response for a multipath channel;

[0028]FIG. 12 is a block diagram of a method for generating correlated tap
coefficients in a T-spaced model, in accordance with an embodiment of the
present invention;

[0029]FIG. 13 is a schematic representation of the spatial pressure
gradient to be measured in accordance with an embodiment of the present
invention;

[0030]FIG. 14 is a graphical illustration of the signal constellation of
the output signal from the Zero Forcing (ZF) equalizer, in accordance
with an embodiment of the invention;

[0031]FIG. 15 depicts an exemplary distribution of receiver locations in
accordance with an embodiment of the invention;

[0032]FIG. 16 is an AC T (Acoustic Toolbox) Bellhop model of channel
impulse responses, using the illustrated simulation parameters, in
accordance with one embodiment of the invention;

[0033]FIG. 17 is a graph of the average pressure impulse response over 16
receiver locations for the initial receiver range of 5 Kilometers (km)
and with a coarse split bottom profile;

[0034]FIG. 18 is a graph of the pressure impulse response at receiver
location 44 for the initial receiver range, using a coarse split bottom
profile;

[0035]FIG. 19 is a graph of the average horizontal velocity impulse
response over 16 receiver locations for the initial receiver range of 5
km and with a coarse silt bottom profile;

[0036]FIG. 20 is a graph of the horizontal velocity impulse response at
receiver location 44 for the initial receiver range of 5 km and a coarse
silt bottom profile;

[0037]FIG. 21 is a graph of the average vertical velocity impulse response
over 16 receiver locations for the initial receiver range of 5 km and
coarse silt bottom profile;

[0038]FIG. 22 is a graph of the vertical velocity impulse response at
receiver location 44 for the initial receiver range of 5 km and a coarse
silt bottom profile;

[0039]FIG. 23 is a graph of the average frequency response of the pressure
impulse response over 16 receiver locations for the initial receiver
range of 5 km and a coarse silt bottom profile;

[0040]FIG. 24 is a graph of the frequency response of the pressure impulse
response at receiver location 44 for the initial receiver range of 5 km
and with a coarse silt bottom profile;

[0041]FIG. 25 is a graph of the average frequency response of the
horizontal velocity impulse response over 16 receiver locations for the
initial receiver range of 5 km and with a coarse silt bottom profile;

[0042]FIG. 26 is a graph of the frequency response of the horizontal
velocity impulse response at receiver location 44 for the initial
receiver range of 5 km and with a coarse silt bottom profile;

[0043]FIG. 27 is a graph of the average frequency response of the vertical
velocity impulse response over 16 receiver locations for the initial
receiver range of 5 km and with a coarse silt bottom profile;

[0044]FIG. 28 is a graph of the frequency response of the vertical
velocity impulse response at receiver location 44 for the initial
receiver range of 5 km and with a coarse silt bottom profile;

[0045]FIG. 29 is a graph of the average bit error rate over 16 receiver
locations for the initial receiver range of 5 km and with a coarse silt
profile;

[0046]FIG. 30 is a graph of the bit error rate at the receiver location 44
for the initial receiver range of 5 km and with a coarse silt profile;

[0047]FIG. 31 is a graph of the normalized average eigen-values over 16
receiver locations for the initial receiver range of 5 km and with a
coarse silt profile;

[0048]FIG. 32 is a graph of the normalized eigen-values at receiver
location 44 for the initial receiver range of 5 km and with a coarse silt
profile;

[0049]FIG. 33 is a graph of the average of inverted diagonal elements of
(HH H)-1 over 16 receiver locations for the initial receiver
range of 5 km and with a coarse silt profile;

[0050]FIG. 34 is a graph of the inverted diagonal elements of (HH
H)-1 at receiver location 44 for the initial receiver range of 5 km
and with a coarse silt profile;

[0051]FIG. 35 is a graph of the average pressure impulse response over 16
receiver locations for an initial receiver range of 10 km and with a
coarse silt bottom profile;

[0052]FIG. 36 is a graph of the pressure impulse response at receiver
location 44 for an initial receiver range of 10 km and with a coarse silt
bottom profile;

[0053]FIG. 37 is a graph of the average horizontal velocity impulse
response over 16 receiver locations for the initial receiver range of 10
km and with a coarse silt bottom profile;

[0054]FIG. 38 is a graph of the horizontal velocity impulse response at
the receiver location 44 for the initial receiver range of 10 km and with
a coarse silt bottom profile;

[0055]FIG. 39 is a graph of the average vertical velocity impulse response
over 16 receiver locations for the initial receiver range of 10 km and
with a coarse silt bottom profile;

[0056]FIG. 40 is a graph of the vertical velocity impulse response at the
receiver location 44 for the initial receiver range of 10 km and with a
coarse silt bottom profile;

[0057]FIG. 41 is a graph of the average frequency response of the pressure
impulse response over 16 receiver locations for the initial receiver
range of 10 km and with a coarse silt bottom profile;

[0058]FIG. 42 is a graph of the frequency response of the pressure impulse
response at the receiver location 44 for the initial receiver range of 10
km and with a coarse silt bottom profile;

[0059]FIG. 43 is a graph of the average frequency response of the
horizontal velocity impulse response over 16 receiver locations for the
initial receiver range of 10 km and with a coarse silt bottom profile;

[0060]FIG. 44 is a graph of the frequency response of the horizontal
velocity impulse response at receiver location 44 for the initial
receiver range of 10 km and with a coarse silt bottom profile;

[0061]FIG. 45 is a graph of the average frequency response of the vertical
velocity impulse response over 16 receiver locations for the initial
receiver range of 10 km and with a coarse silt bottom profile;

[0062]FIG. 46 is a graph of the frequency response of the vertical
velocity impulse response at receiver location 44 for the initial
receiver range of 10 km and with a coarse silt bottom profile;

[0063]FIG. 47 is a graph of the average bit error rate over 16 receiver
locations for the initial receiver range of 10 km and with a coarse silt
profile;

[0064]FIG. 48 is a graph of the bit error rate at receiver location 44 for
the initial receiver range of 10 km and with a coarse silt profile;

[0065]FIG. 49 is a graph of the normalized average eigenvalues over 16
receiver locations for the initial receiver range of 10 km and with a
coarse silt profile;

[0066]FIG. 50 is a graph of the normalized eigenvalues at the receiver
location 44 for the initial receiver range of 10 km and with a coarse
silt profile;

[0067]FIG. 51 is a graph of the average of the inverted diagonal elements
of (HH H)-1 over 16 receiver locations for the initial receiver
range of 10 km and with a coarse silt profile;

[0068]FIG. 52 is a graph of the inverted diagonal elements of (HH
H)-1 at the receiver location 44 for the initial receiver range of
10 km and with a coarse silt profile;

[0069]FIG. 53 is a graph of the average pressure impulse response over 16
receiver locations for the initial receiver range of 5 km and with very
fine sand bottom profile;

[0070]FIG. 54 is a graph of the pressure impulse response at the receiver
location 44 for the initial receiver range of 5 km and with a very fine
sand bottom profile;

[0071]FIG. 55 is a graph of the average horizontal velocity impulse
response over 16 receiver locations for the initial receiver range of 5
km and with a very fine sand bottom profile;

[0072]FIG. 56 is a graph of the horizontal velocity impulse response at
the receiver location 44 for the initial receiver range of 5 km and with
a very fine sand bottom profile;

[0073]FIG. 57 is a graph of the average vertical velocity impulse response
over 16 receiver locations for the initial receiver range of 5 km and
with a very fine sand bottom profile;

[0074]FIG. 58 is a graph of the horizontal velocity impulse response at
receiver location 44 for the initial receiver range of 5 km and with a
very fine sand bottom profile;

[0075]FIG. 59 is a graph of the average frequency response of the pressure
impulse response over 16 receiver locations for the initial receiver
range of 5 km and with a very fine sand bottom profile;

[0076]FIG. 60 is a graph of the frequency response of the pressure impulse
response at receiver location 44 for the initial receiver range of 5 km
and with a very fine sand bottom profile;

[0077]FIG. 61 is a graph of the average frequency response of the
horizontal velocity impulse response over 16 receiver locations for the
initial receiver range of 5 km and with a very fine sand bottom profile;

[0078]FIG. 62 is a graph of the frequency response of the horizontal
velocity impulse response at receiver location 44 for the initial
receiver range of 5 km and with a very fine sand bottom profile;

[0079]FIG. 63 is a graph of the average frequency response of the vertical
velocity impulse response over 16 receiver locations for the initial
receiver range of 5 km and with a very fine sand bottom profile;

[0080]FIG. 64 is a graph of the frequency response of the vertical
velocity impulse response at receiver location 44 for the initial
receiver range of 5 km and with a very fine sand bottom profile;

[0081]FIG. 65 is a graph of the average bit error rate over 16 receiver
locations for the initial receiver range of 5 km and with a very fine
sand bottom profile;

[0082]FIG. 66 is a graph of the bit error rate at receiver location 44 for
the initial receiver range of 5 km and with a very fine sand bottom
profile;

[0083]FIG. 67 is a graph of the normalized average eigenvalues over 16
receiver locations for the initial receiver range of 5 km and with a very
fine sand bottom profile;

[0084]FIG. 68 is a graph of normalized eigenvalues at receiver location 44
for the initial receiver range of 5 km and with a very fine sand bottom
profile;

[0085]FIG. 69 is a graph of the average of inverted diagonal elements of
(HH H)-1 over 16 receiver locations for the initial receiver
range of 5 km and with a very fine sand bottom profile;

[0086]FIG. 70 is a graph of the inverted diagonal elements of (HH
H)-1 at receiver location 44 for the initial receiver range of 5 km
and with a very fine sand bottom profile;

[0087]FIG. 71 is a graph of the average pressure impulse response over 16
receiver locations for the initial receiver range of 10 km and with a
very fine sand bottom profile;

[0088]FIG. 72 is a graph of the pressure impulse response at receiver
location 44 for the initial receiver range of 10 km and with a very fine
sand bottom profile;

[0089]FIG. 73 is a graph of the average horizontal velocity impulse
response over 16 receiver locations for the initial receiver range of 10
km and with a very fine sand bottom profile;

[0090]FIG. 74 is a graph of the horizontal velocity impulse response at
receiver location 44 for the initial receiver range of 10 km and with
very a fine sand bottom profile;

[0091]FIG. 75 is a graph of the average vertical velocity impulse response
over 16 receiver locations for the initial receiver range of 10 km and
with a very fine sand bottom profile;

[0092]FIG. 76 is a graph of the vertical velocity impulse response at
receiver location 44 for the initial receiver range of 10 km and with a
very fine sand bottom profile;

[0093]FIG. 77 is a graph of the average frequency response of the pressure
impulse response over 16 receiver locations for the initial receiver
range of 10 km and with a very fine sand bottom profile;

[0094]FIG. 78 is a graph of the frequency response of the pressure impulse
response at receiver location 44 for the initial receiver range of 10 km
and with a very fine sand bottom profile;

[0095]FIG. 79 is a graph of the average frequency response of the
horizontal velocity impulse response over 16 receiver locations for the
initial receiver range of 10 km and with a very fine sand bottom profile;

[0096]FIG. 80 is a graph of the frequency response of the horizontal
velocity impulse response at receiver location 44 for the initial
receiver range of 10 km and with a very fine sand bottom profile;

[0097]FIG. 81 is a graph of the average frequency response of the vertical
velocity impulse response over 16 receiver locations for the initial
receiver range of 10 km and with a very fine sand bottom profile;

[0098]FIG. 82 is a graph of the frequency response of the vertical
velocity impulse response at receiver location 44 for the initial
receiver range of 10 km and with a very fine sand bottom profile;

[0099]FIG. 83 is a graph of the average bit error rate over 16 receiver
locations for the initial receiver range of 10 km and with a very fine
sand bottom profile;

[0100]FIG. 84 is a graph of the bit error rate at receiver location 44 for
the initial receiver range of 10 km and with a very fine sand bottom
profile;

[0101]FIG. 85 is a graph of the normalized average eigen-values over 16
receiver locations for the initial receiver range of 10 km and with a
very fine sand bottom profile;

[0102]FIG. 86 is a graph of the normalized eigen-values at receiver
location 44 for the initial receiver range of 10 km and with a very fine
sand bottom profile;

[0103]FIG. 87 is a graph of the average of inverted diagonal elements of
(HH H)-1 over 16 receiver locations for the initial receiver
range of 10 km and with a very fine sand bottom profile; and

[0104]FIG. 88 is a graph of the inverted diagonal elements of (HH
H)-1 at receiver location 44 for the initial receiver range of 10 km
and with very fine sand bottom profile.

DETAILED DESCRIPTION OF THE INVENTION

[0105]Acoustic fields may be used to convey data. A source may be used to
encode a package of data in an acoustic field. Sources may include any
device, system, or method capable of converting a data package into an
acoustic field, such as a transmitter. A receiver may be used to receive
the acoustic field. Receivers may include any device, system, or method
capable of receiving an acoustic field. Receiving the acoustic field may
include measuring one or more parameters of the acoustic field, such as
one or more vector components and/or a scalar component. The values of
the measured parameters may be used to recover the data package conveyed.
Some embodiments utilize measurements of one or more parameters of the
acoustic field to recover the conveyed information.

[0106]Acoustic fields may be used to convey data in any environment
including liquids, gases, solids and/or any combinations thereof. Herein,
"fluid" may include liquid and/or gas. For example, acoustic fields may
be used to transmit data in underwater channels. Data capable of being
conveyed include, but are not limited to, any information, which may be
encoded in an acoustic field. For example, a transducer may be used to
convert a data package (e.g., electrical signal) into a pressure field,
and a receiver may reconvert the pressure signals back into electrical
waveforms.

[0107]An embodiment includes measuring any quantifiable parameter of an
acoustic field to determine the data conveyed. In an embodiment, one or
more vector components of an acoustic field, in addition to the scalar
component (i.e., pressure), may be measured to determine values for the
data transmitted. Vector components of an acoustic field include, but are
not limited to, the three components of the acoustic particle velocity
(i.e., the pressure gradients or the spatial derivatives of the particle
displacement), any of the nine components of the spatial derivative of
the acoustic particle velocity (i.e., the velocity gradients), and any
higher-order gradients of the acoustic field. For example, in a
three-dimensional underwater channel, x, y, and z components of the
acoustic particle velocity may be measured, as well as nine components of
the velocity gradient.

[0108]In an embodiment, one vector component may be measured to discern
values for the conveyed data. Alternately, multiple vector components may
be measured to discern values for the conveyed data. For example, an
embodiment may include measuring pressure, components of acoustic
particle velocity, and components of acoustic particle velocity
gradients. In this example, thirteen channels would be available for the
conveyance of data. In some embodiments, measurement of multiple acoustic
parameters to determine values for conveyed data may decrease the error
probability of data recovery while utilizing a small array. In other
words, use of both scalar and vector components of the acoustic field
increases the number of channels available for conveyance of data, and
thereby decreases the error probability.

[0109]Vector components of an acoustic field may be measured using devices
including, but not limited to, transducers, receivers, receivers, vector
sensors (e.g., inertial sensors, gradient sensors, uniaxial vector
sensors, biaxial vector sensors, and/or triaxial vector sensors),
multi-axial vector sensors, higher order sensors (e.g., dyadic or tensor
sensors), accelerometers (e.g., uniaxial accelerometers), hydrophones,
fiber optic-based sensors, or any other devices known in the art or yet
to be developed that achieve the same or similar functionality.
Measurements of the scalar components of the acoustic field may be made
using devices which include, but are not limited to, pressure sensors,
transducers, hydrophones, omni-directional hydrophones, directional
hydrophones and/or any other devices known in the art or yet to be
developed that achieve the same or similar functionality. Recovering
information from the vector components of the acoustic field is not
limited to any particular sensor type, any device capable of measuring a
vector component of the acoustic field suffices.

[0110]In an embodiment, a signal may be processed at a receiver using one
or multiple processing methods. Processing methods may include any signal
processing methods known in the art or yet to be developed that achieve
the same or similar functionality, such as equalization algorithms,
diversity techniques, decoding methods, interference cancellation
techniques, temporal and frequency processing, etc. For example, any
known or yet to be developed digital and/or analog signal processing
method may be used in an embodiment. Although a single-user communication
system and method is discussed throughout this application, the inventive
principles discussed herein are fully applicable to multi-user
communication systems and networks.

[0111]Algorithms utilized may include, but are not limited to, different
types of single and multi-channels equalizers such as zero-forcing
equalizer, a minimum mean square equalizer (herein referred to as MMSE),
a decision-feedback equalizer, adaptive equalizers and turbo-equalizers
with different types of training algorithms, and/or any processing
algorithm used in the art or yet to be developed that achieve the same or
similar functionality. In addition to these temporal equalization
algorithms, space-time and space-frequency techniques may be used as
well.

[0112]In some embodiments, one or more diversity techniques may be used to
combine the measured components including, but not limited to maximal
ratio combiner, selection combiner, equal gain combiner, and/or any other
techniques known in the art or yet to be developed that achieve the same
or similar functionality.

[0113]An embodiment includes utilizing a decoding method determined by the
code used at the source. The codes used at the source may be source
coding (e.g., data compression) channel coding (e.g., temporal codes,
space-time codes, space-time-frequency codes), joint source-channel codes
and/or any other methods known in the art or yet to be developed that
achieve the same or similar functionality. Further, some embodiments
include a processing method capable of performing carrier and/or bit
synchronization.

[0114]Some embodiments include processing methods occurring at the
receiver. In alternate embodiments, one or more processing devices may be
positioned proximate to the receiving device. For example, with an
Orthogonal Frequency Division Multiplexing (herein referred to as OFDM)
signal transmitted, fast Fourier transform (herein referred to as FFT)
blocks are needed at the receiver. In another example, transmitting a
spread spectrum signal (e.g., code division multiple access, direct
sequence, or frequency hopping) for low-probability of interception
communication and/or multi-user communication may create a need for a
dispreading module at the receiver for certain embodiments.

[0115]In some embodiments, a combination of receivers may be used to
measure the acoustic field. For example, a vector sensor may be used in
combination with a hydrophone to measure all the acoustic field
components. The acoustic field components are used to determine data
values for the conveyed data.

[0116]An embodiment may include commercially available vector sensors used
as receivers. Alternate embodiments may include using a vector sensor in
a transceiver to aid in relaying signals.

[0117]In one embodiment, an inertial vector sensor is used to measure the
velocity or acceleration by responding to acoustic particle motion. In
alternate embodiments, gradient sensors may be used which utilize a
finite-difference approximation to estimate gradients of the acoustic
field such as velocity and acceleration.

[0118]In some embodiments, a vector sensor may have the capacity to
measure multiple parameters of the acoustic field. A vector sensor may be
designed to measure the scalar component of the acoustic field, as well
as multiple vector components of the acoustic field, simultaneously.

[0119]Receivers may be arranged and/or designed to eliminate a need for
arrays of pressure-only receiving devices. For example, use of a system
of vector sensors may eliminate a need for large-size pressure-only
arrays. In some embodiments, vector sensors may be used as compact
multi-channel receivers, measuring both the scalar and vector components
in a single point in space. In contrast are the conventional systems such
as pressure-only sensors spatially separated and arranged in large size
arrays. Thus, the volume of space required for the receivers and/or
decoding devices may be greatly reduced. The decreased size of the
receivers and/or decoding devices make the technology available for a
wider variety of applications which were previously prohibited. Some of
examples of this include, but are not limited to, small autonomous
underwater vehicles, divers communicating with each other and a
submarine, etc.

[0120]In some embodiments, devices used to measure acoustic wave
parameters may be neutrally buoyant in the fluid through which the
acoustic field is traveling. For example, vector sensors may be neutrally
buoyant in a fluid such as water.

[0121]An acoustic field communication system may include single input
single output systems (herein referred to as SISOs), single input
multiple output systems (herein referred to as SIMOs), multiple input
single output systems (herein referred to as MISOs), and multiple input
multiple output systems (herein referred to as MIMOs).

[0122]Conveyed data may include voice, video, text, numbers, characters,
images, control and command signals, telemetry signals, and/or other
outputs from devices used to convert physical quantities into data
communication symbols.

[0123]Signals transmitted via the acoustic field communication system may
be modulated. In some embodiments, modulation may include, but is not
limited to, angular modulation, phase modulation (herein referred to as
PM), frequency modulation (herein referred to as FM), amplitude
modulation (herein referred to as AM), single-sideband modulation (herein
referred to as SSB), single-sideband suppressed carrier modulation
(herein referred to as SSB-SC), vestigial-sideband modulation (herein
referred to as VSB), sigma-delta modulation, phase-shift keying (herein
referred to as PSK), frequency-shift keying (herein referred to as FSK),
audio frequency-shift keying (herein referred to as AFSK), minimum-shift
keying (herein referred to as MSK), Gaussian minimum-shift keying (herein
referred to as GMSK), very minimum-shift keying (herein referred to as
VMSK), binary phase-shift keying (herein referred to as BPSK), quadrature
phase-shift keying (herein referred to as QPSK), offset or staggered
phase-shift keying (herein referred to as SQPSK), π/4 quadrature
phase-shift keying (herein referred to as π/4 QPSK), differential
phase-shift keying (herein referred to as DPSK), amplitude-shift keying
(herein referred to as ASK), on-off keying (herein referred to as OOK),
quadrature amplitude modulation (herein referred to as QAM), continuous
phase modulation (herein referred to as CPM), trellis coded modulation
(herein referred to as TCM), polar modulation, pulse-code modulation,
pulse-width modulation, pulse-amplitude modulation, pulse-position
modulation, pulse-density modulation, space-time modulations (e.g.,
unitary, rotated constellation), multi-carrier methods such as OFDM, and
any other modulation systems known in the art or yet to be developed that
achieve the same or similar functionality.

[0124]An embodiment of the system increases the number of channels for
data communication by utilizing vector components of an underwater
acoustic field. Further, the system optimizes the use of the bandwidth
available. For example, bandwidth available for use in underwater
environments may be a limiting factor. In an embodiment, a volume of
space required for an acoustic field communication system may be reduced
by utilizing receivers using vector components of the acoustic field to
determine the content of conveyed data.

[0125]Use of receivers capable of determining vector components of an
acoustic field may, in some embodiments, increase a rate of data transfer
and increase the reliability of the communication system.

[0126]In an embodiment, a basic set of equations for data detection
utilizing vector sensors is derived. A simple set of equations is used to
demonstrate the fundamental idea of how both the vector and scalar
components of the acoustic field can be-utilized for data reception. In
an embodiment, two pressure sources transmit the data, and a receiver,
here a vector sensor, measures the pressure and one component of the
particle velocity. This is basically a 2×2 multiple-input
multiple-output (MIMO) system. In some embodiments, a vector sensor
capable of measuring more components of the acoustic particle velocity
may be used. Certain embodiments may include arrays of spatially
separated vector sensors. A vector sensor embodiment may work as a
receiver with only one pressure sensor. In some embodiments, there may be
one or multiple pressure sources transmitting data. An embodiment using
two or more pressure sources, and at least one vector sensor receiver,
may realize the numerous advantages of MIMO communication systems.

[0127]As shown in FIG. 1, one embodiment includes two pressure sources
S1 and S2 and one vector sensor R1 that measures the pressure
and the z-component of the acoustic particle velocity. The black dots at
the transmitting device represent two pressure sources (transmitters),
S1 and S2, whereas the single black square at the receiver
R1 represents a vector sensor. In an embodiment, the vector sensor
may include an inertial or a gradient sensor. For example, a PMN-PT (lead
magnesium niobate-lead titanate)-based accelerometer may be used. In
alternate embodiments, the black square in FIG. 1 represents a device
having the capability to measure the pressure and the z-component of the
acoustic particle velocity at a single point in space, either by truly
measuring the acoustic particle velocity or approximating it using a
finite-difference operation. In other embodiments, receiver R1 may
measure one or more of pressure, velocity in plurality of directions,
such as orthogonal X, Y, and Z directions, and/or velocity gradients in
any of three directions for any of the plurality of direction-specific
velocity components (i.e. Vx, Vy, and/or Vz) of the fluid in the vicinity
of receiver R1. Thus, in some cases, up to thirteen quantities may be
available for measurement (one pressure quantity, three velocity
quantities, and/or nine velocity gradient measurements).

[0128]Equations to define the operation of the communication system are
described below. Equations derived may apply to both gradient and
inertial sensors. FIG. 1 depicts two pressure sources (T1, and T2) used
to transmit symbols s1 and s2. The pressure sources T1 and T2
are located at the depths z1 and z2, respectively, such that
z2>z1≧0. The vector sensor receiver, R1, is located
at the depth z1, at a distance from the two pressure sources. In
some embodiments, one or more sources and one or more receivers may be
positioned at the same depth. Alternately, sources and receivers may be
positioned at varying depths.

[0129]A channel between the pressure sources and the receiver includes two
pressure channel coefficients p11 and p12, represented by
straight dashed lines in FIG. 1. There are also two pressure-equivalent
velocity channel coefficients p11z and p12z in FIG.
1, represented by curved dashed lines. To define p11z and
p12z, we define the two velocity channel coefficients
v11z and v12z. According to the linearized momentum
equation or Euler's equation, the z-component of the velocity at location
z1 of the receive side and at the frequency f0, due to the
pressure sensor at z1 of the transmit side is given by:

v 11 z = 1 j ρ 0 ω 0 ∂
p 11 ∂ z . ( 1 )

[0130]In the above equation, ρ0 is defined as the density of the
fluid, j2=-1, and the frequency having units of rad/sec is defined
as ω0=2πf0. Eq. (1) simply states that the velocity in
a certain direction is proportional to the spatial pressure gradient in
that direction. In this example, the z-component of the acoustic particle
velocity is discussed, however, in a three-dimensional underwater
channel, x and y components of the acoustic particle velocity can be
measured, as well as nine components of the velocity gradient, etc.

[0131]To simplify the notation, multiply the velocity channel coefficient
in Eq. (1) by -ρ0c, the negative of the acoustic impedance of
the fluid, where c is the speed of sound. This gives the associated
pressure-equivalent velocity channel coefficient as
p11z=-ρ0cv11z. Defining the wavelength as
λ and the wave number as k=2π/λ=ω0/c the
following equation is obtained:

p 11 z = 1 j k ∂ p 11
∂ z . ( 2 )

[0132]Similarly,
v12z=-(jρ0ω0)-1∂p12/-
∂z and
p12z=-ρ0cv12z=(jk)-1∂p.sub-
.12/∂z are derived. In a time-invariant environment with no
multipath, all the four channel coefficients p11, p12,
p11z and p12z are constant complex numbers.

[0133]In FIG. 1 additive ambient noise pressure on the receiver side at
z1, is represented by n1. At the same location, the z-component
of the additive ambient noise velocity sensed by the vector sensor is
given by η1z=-(jρ0ω0)-1.differenti-
al.n1/∂z, derived in the same manner as Eq. (1). This is
the vertical spatial gradient of the noise pressure at z1 on the
receiver side. In FIG. 1, the associated vertical pressure-equivalent
additive ambient noise velocity on the receiver side, at location
z1, is given by
n1z=-ρ0cη1z=(jk)-1∂n.s-
ub.1/∂z.

[0134]According to FIG. 1, the received pressure signal at z1 can be
written as r1=p11s1+p12s2+n1. The
z-component of the pressure-equivalent received velocity signal at
z1 is defined as
r1z=(jk)-1∂r1/∂z. By taking
the spatial gradient of r1 with respect to z and according to Eq.
(2), as well as
p12z=(jk)-1∂p12/∂z and
n1z=(jk)-1∂n1/∂z, results
in the following equation
r1z=p11zs1+p12zs2+n1z.
Therefore, the two key equations for the proposed MIMO system in FIG. 1,
having two inputs and two outputs, can be summarized as:

r1=p11s1+p12s2+n1,

r1z=p11zs1+p12zs2+n1z
(3)

[0135]In this embodiment, the two output signals r1 and r1z
are measured at a single point in space. This illustrates the possibility
of having a two channel compact receiver, without two spatially separated
receive pressure sensors.

[0136]In what follows, Eq. (3) is used to demonstrate how the two basic
gains of a MIMO system (i.e., diversity and multiplexing gains) can be
achieved in the proposed vector sensor system.

[0137]In an embodiment having one source, data may be transmitted
sequentially, over 2T sec., where T is the symbol duration. Thus, there
is no multiplexing gain. Embodiments using multiple sources as shown in
FIG. 1, allow the symbols to be transmitted simultaneously, which takes
only T sec. This results in a multiplexing gain on the order of two.

[0138]To recover s1 and s2 at the receiver, one can solve the
set of linear equations in Eq. (3), using well-known standard methods
such as zero forcing (herein referred to as ZF), minimum mean square
error (herein referred to as MMSE), maximum likelihood (herein referred
to as ML) symbol detectors, or any other method known in the art or yet
to be developed that achieves the same or similar functionality. Sending
training symbols prior to data transmission, allows accurate estimates of
the channel components in Eq. (3), (i.e., p11, p12,
p11z and p12z) to be obtained at the receiver.
Overall, by using two pressure sources, T1 and T2, and a small size
vector sensor receiver, R1, measuring the pressure and the z-component of
the acoustic velocity, the transmission rate is doubled.

[0139]In an embodiment, diversity decreases the symbol error probability.
Diversity is defined as symbol reception utilizing more than one channel.
Thus, if one channel is in deep fade such that the transmitted symbol is
destroyed, the transmitted symbol may be recovered from other channels
which are still in good condition. This feature may also be referred to
as "redundancy", which may be achieved by receiving the same transmission
at multiple, such as four, reception channels. If one channel fails, the
other three channels may still recover the transmitted data. In general,
an increase in the number of available channels, increases the likelihood
of transmitted data being successfully received, even under adverse
conditions in which one or more channels are inoperative.

[0140]In some embodiments, a diversity gain of second order may be
obtained via the vector sensor receiver R1 in FIG. 1. Suppose s1 is
transmitted from the first pressure source T1 and s2=0, i.e., no
transmission from the second pressure source. This is basically a
1×2 single-input multiple-output (SIMO) system. According to Eq.
(3), the system equations can be written as

r1=p11s1+n1,

r1z=p11zs1+n1z. (4)

[0141]Clearly, Eq. (4) shows the transmitted symbol s1 is received
via two different channels p11 and p11z. If the two
channels are uncorrelated, then this simply means a receiver diversity
gain of second order, obtained by taking advantage of the z-component of
the acoustic velocity, as well as pressure, with a small size vector
sensor. In some embodiments, certain conditions (e.g., estimates of the
channel coefficients) may influence the choice of the optimal receiver.
In an embodiment, the optimal receiver may be the maximal ratio combiner
(MRC) i.e., p*11r1+{p11z}*r1z, where * is
the complex conjugate. In some embodiments, the pressure/velocity
receiver diversity scheme with a vector sensor differs from the pressure
spatial diversity, obtained by widely-spaced pressure sensors arranged in
large arrays.

[0142]For some embodiments, the correlation among the pressure and
acoustic particle velocity components plays a key role in achieving
multiplexing and diversity gains in the vector sensor system shown in
FIG. 1. On one hand, the rank of the 2×2 channel matrix in Eq. (3),
which is related to the correlation between the pressure and velocity
components, directly affects the performance of the 2×2 MIMO system
that offered a multiplexing gain. On the other hand, the correlation
between p11 and p11z in Eq. (4) determines the performance
of the 1×2 SIMO system, which had a diversity gain.

[0143]In an embodiment, n1 and n1z are uncorrelated in
cases of practical interest, i.e., E[n1{n1z}*]=0.
Furthermore, they have different powers in general, which means
E[|n1|2]≠E[|n1z|2]. To calculate
E[n1{n1z}*], let us define n(z) as the ambient noise
pressure at the receiver side and a depth, z. Similarly, n'(z) is the
vertical spatial derivative of n(z) at the same location, i.e.,
n'(z)=∂n(z)/∂z. Therefore the noise components
in Eq. (3) can be written as n1=n(z1) and
n1z=(jk)-1n'(z1). In some embodiments, n(z) is
defined as a complex zero-mean unit-power noise, i.e., E[n(z)]=0 and
E[|n(z)|2]=1, where E is the expectation operator. Further, for some
embodiments the following two identities are obtained:

E [ n ( z + l ) { n ' ( z ) } ' ] = -
∂ q n ( l ) ∂ l , ( 5 )

E [ n ' ( z + l ) { n ' ( z ) } * ] =
- ∂ 2 q n ( l ) ∂ l 2 .
( 6 )

[0144]For Eqs. (5) and (6), qn(l) is the vertical spatial correlation
of the ambient noise pressure, defined by qn(l)=E[n(z+l)n*(z)].

[0145]For some embodiments, the correlation of interest between the noise
components in Eq. (3) is written as
E[n1{n1z}*]=(-jk)-1E[n(z1){n'(z1)}*]. When
z=z1 and l=0 in Eq. (5) the following relationship is obtained
E[n1{n1z}*]=(jk)-1∂qn(l)/.different-
ial.l|l=0. If qn(l) is real, its Fourier transform has even
symmetry. Then using the Fourier transform of the derivative of
qn(l), it is easy to verify that
∂qn(l)/∂l|l=0=0. For embodiments
under rather general conditions E[n1{n1z}*]=0, i.e., the
two noise components in Eq. (3) are uncorrelated.

[0146]There are several commonly-used ambient noise models for which
∂qn(l)/∂l|l=0=0. Utilized noise
models may include, but are not limited to, a three-dimensional (3D)
isotropic noise model, a two-dimensional (2D) isotropic noise model, also
known as the impulsive noise model, and the surface generated noise
model. In the first model the angular noise distribution at the receiver
is isotropic in the 3D volume, whereas in the second one the isotropic
angular distribution of the noise at the receiver is restricted to the 2D
y-z plane in FIG. 1. In an embodiment of the 3D model
qn(l)=sin(kl)/(kl). An alternate embodiment using the 2D model
defines qn(l)=J0(kl), and J0(.) as a zero-order Bessel
function of the first kind. Some embodiments include a vertical spatial
correlation for the surface generated noise defined as
qn(l)=2J1(kl)/(kl), where J1(.)is the first-order Bessel
function of the first kind. Note that qn(l) in all these models is
real. So, as explained before, ∂qn(l)/∂l
is zero at l=0. This makes the two noise components in the proposed
vector sensor receiver in FIG. 1 uncorrelated.

[0147]In some embodiments, E[|n1z|2], is calculated
assuming E[|n1|2]=1. Using n1z=(jk)-1n'(z1)
the following expression is obtained
E[|n1z|2]=k-2E[|n'(z1)|2]=-k-2.differe-
ntial.2qn(l)/∂l2|l=0, where the last
identity is derived from equation (6) when l=0.

[0148]In some embodiments, where qn(l)=sin(kl)/(kl), J0(kl) and
2J1(kl)/(kl), respectively, it can be shown that
E[|n1z|2]=1/3, 1/2, and 1/4, respectively. Some
embodiments of the noise models represent unit-power pressure noises
because E[|n1|2]=qn(0)=1. Thus, in some embodiments, noise
components of the vector sensor receiver in FIG. 1 have different powers.
An embodiment includes a vector sensor that measures the acoustic
particle velocity has a directional pattern, compared to the
omni-directional pressure meter in that vector sensor.

[0149]In some embodiments, large-size pressure-only array receivers may be
replaced by compact vector sensors, while providing the same level of
system performance, in terms of symbol error rate (herein referred to as
SER) and channel capacity.

[0150]In an acoustic communication system embodiment, a coherent binary
phase shift keying (BPSK) system may be used in an underwater flat
Rayleigh fading channel. In some embodiments having one pressure sensor
at the source and one pressure sensor at the receiver, the SER may have a
value of about 0.02 at a signal-to-noise ratio (herein referred to as
SNR) of about 10 dB. This SER value may reduce to a value of about 0.006
at the same SNR, if two vertically well-separated pressure sensors are
used at the receiver. Depending on the spatial coherence of the field and
the carrier frequency, the element spacing at the receiver might be
large. For example, in an embodiment with carrier frequency equal to
about 1.2 kHz, the vertical spatial correlation is negligible if the two
pressure sensors are spaced by at least 5 m.

[0151]Some embodiments include one pressure sensor at the transmitter and
one vector sensor at the receiver measuring both the pressure and the
vertical component of the velocity. In a receiver embodiment, the same
low SER is achievable by the vector sensor as a pressure sensor array.
The size of the vector sensor receiver, which senses the pressure and
particle velocity in a single point in space could be much smaller than a
pressure-only array.

[0152]Embodiments utilizing higher order sensors (e.g., dyadic or tensor
sensors) measure additional components of the field such as velocity
gradients (e.g., second-order spatial gradients of the pressure). Using
such sensors, one might be able to obtain higher order diversity gains
with a small receiver. For example, acceleration (i.e., the spatial
gradient of the velocity) has nine components, whereas the velocity
(i.e., spatial gradient of pressure) has three components.

[0153]In some embodiments, employing multiple transmitters and receivers
in multipath channels, allows for several separate spatial channels
between the transmitter and receiver. This results in much higher channel
capacities.

[0154]In an embodiment having one pressure sensor as the transmitter and
another pressure sensor as the receiver, a channel capacity in a Rayleigh
fading channel approximately equals B log2(1+SNR) bits/sec, where B
is the channel bandwidth in Hz and log2(.) is the base-2 logarithm.
In some embodiments, this channel capacity may represent the maximum data
rate that one can send through the channel. At data rates higher than
this limit, a rate of errors in the transmission increases. For example,
with a channel bandwidth of 5 kHz (i.e., B=5 kHz), the channel capacity
is approximately 30 kbits/sec, when SNR=20 dB. In an alternate
embodiment, a system has two pressure sensors at the source and two
pressure sensors at the receiver in a multipath channel. If the element
spacing in both transmit and receive arrays is large enough, then the
capacity is almost two times larger, i.e., 2×B log2(1+SNR),
which results in a 60 kbits/sec capacity in our numerical example.

[0155]An embodiment of an acoustic communication system including vector
sensors at the receiver may have a channel capacity similar to a system
having a pressure-sensor array at the receiver. Based on the system
equations derived herein for a system having two pressure sensors at the
source and one vector sensor at the receiver, the channel capacity is
almost the same as 60 kbits/sec, but with a more compact receiver.

[0156]A simulation was used to predict the behavior of a multi-channel
underwater communication receiver with a vector sensor. Additional
simulations utilizing alternative underwater channels are included in
Appendix A. In the simulation, a shallow-water channel was simulated at
fc=300 Hz. The channel depth was set to 58 m and the distance
between the source and the receiver was set to 10 km. Using narrowband
phase-shift keying BPSK transmission from a single pressure sensor, the
model simulated the bit error probability of three systems: (A) a
pressure-only system with one receive sensor, (B) a pressure-only system
with two widely-spaced receive sensors, vertically separated by
1.2λ=6m to be uncorrelated, and (C) the new vector-sensor
communication system with a single transmitter and a receive dipole, to
measure the vertical component of the velocity (element
spacing=0.2λ=1 m). From the results of the simulation, system B had
a reduced bit error probability when compared to system A. This may have
been due to the spatial diversity. According to the simulation, system C
showed nearly the same performance as B. For example, to reduce the bit
error probability to a value of about 10-2, the required signal to
noise ratio value (herein referred to as SNR) for systems B and C was 8
and 9 dB, respectively. Thus, use of acoustic particle velocity can be
beneficial to communication. As shown, the compact vector sensor provided
a pressure/velocity diversity gain, similar to the pressure spatial
diversity of a large array (1 m versus 6 m in this example). The
difference between systems B and C may be due to dissimilar noise
characteristics.

Channel Modeling Issues for Vector Sensors

[0157]One benefit for successful deployment a vector-sensor-based
communication system is a good understanding of the propagation
characteristics of the vector components of the field. When compared with
modeling a channel in which scalar sensors are used, channel
characterization for vector sensors confronts different problems. For
example, consider a vector sensor which measures the field pressure, as
well as the three components of the particle velocity, all at a single
point. In this situation, we theoretically have four orthogonal channels.
However, the extent of correlation between these four co-located
channels, especially in the complex multi-path shallow water medium,
which has extensive boundary interactions, needs to be understood. It is
also important to relate these correlations to some key channel
parameters such as delay and angle spreads.

[0158]The shallow water acoustic channel is basically a waveguide, bounded
from bottom and the top. The sea floor is a rough surface which
introduces scattering, reflection loss, and attenuation by sediments,
whereas the sea surface is a rough surface that generates scattering and
reflection loss, attenuation by turbidity and bubbles, and noise due to
surface weather. When compared with deep water acoustic transmission,
shallow water acoustic transmission is more complex, due to the many
interactions of acoustic waves with boundaries, which result in a
significant amount of multi-path propagation. The harsh multi-path
propagation, with delay spreads up to several hundreds of symbols for
high data rates make shallow waters very hard to cope with. High spatial
variability and strong signal fading further complicate the communication
in such channels.

[0159]In this section, first we review the existing techniques for random
underwater modeling. Then we develop a ray-based statistical channel
modeling approach, suitable for the analysis and design of vector sensor
receivers.

Statistical Representations of Pressure and Velocity in Multipath Shallow
Waters

[0160]The available methodologies for studying wave propagation in random
media (WPRM) generally attempt to understand the channel physics and
focus on in-depth analysis of the complex phenomena involved. This
approach becomes more involved when applied to shallow waters.
Application of WPRM methods to the proposed vector sensor systems appear
to yield more complex field representations and correlation expressions.
From a communication engineering point of view, however, such a detailed
analytic WPRM perspective and the resulting complex solutions exceed what
is required by a system designer. For a proper communication system
design, one needs relatively simple yet accurate system-level channel
models, which provide those key pieces of information that directly
affect the system performance.

[0161]Here we develop a new statistical approach, which concentrates on
channel characterization using simple probabilistic models for the random
components of the propagation environment. In this way, the statistical
behavior of the channel can be accurately imitated, and compact
expressions for the correlation functions of interest can be easily
derived, without solving stochastic wave equations. These vector sensor
parametric correlation expressions allow engineers to design, simulate,
and assess a variety of communication schemes under different channel
conditions.

[0162]Consider the Single-Input-Multiple-Output (SIMO) vector sensor
system implemented in a shallow water channel, as shown in FIG. 2. In the
Two-Dimensional (2D) y-z (range-depth) plane, there is one pressure
transmitter at the left of the y field, labeled Tx. There are also two
receive vector sensors at y=0, sensor Rx1 and sensor Rx2, at the depths
z=z1 and z1+L, respectively, with the distance L corresponding
to the spacing between the sensors in the receive vector sensor array.
The center of the array of sensors, Rx1 and Rx2, is considered to be
located at depth z=D.

[0163]Each vector sensor measures the pressure, as well as they and z
components of the particle velocity, all at a single point. This means
that there are two pressure channel coefficients p1 and p2, as
well as four pressure-equivalent velocity channel coefficients
p1y, p1z, p2y and p2z, as shown
in FIG. 2. Each vector sensor can provide three output signals. For
example, Rx1 generates one pressure signal r1 and two
pressure-equivalent velocity signals r1y and r1z,
measured in the y and z directions, respectively. The noises at the
receivers are not shown, for the sake of simplicity. As can be seen, this
is a 1×6 SIMO system. The goal of this subsection is to provide
proper statistical representations for all these pressure and velocity
channels in shallow waters. These channel representations will be used
later in other subsections, to calculate different types of channel
correlations that affect the system performance. Extension of the above
concept to three or more vector sensors is straightforward. Furthermore,
A SIMO system is considered here to specifically focus on channel
modeling issues for vector sensors at the receiver. Generalization of the
results to a Multiple-Input-Multiple-Output (MIMO) setup, where there are
multiple pressure transmitters, can be done in a similar way, when the
SIMO configuration is well understood.

Pressure-Related Channel Functions

[0164]In this subsection we define and focus on the three pressure channel
functions χ(γ,τ), p(τ) and P(f), over the angle-delay,
delay-space and frequency-space domains, respectively.

[0165]FIG. 3 shows the SIMO system of FIG. 2, as well as the geometrical
details of the received rays in a shallow water channel, with two vector
sensor receivers Rx1 and Rx2 (represented by solid black squares in FIG.
3). Two-dimensional propagation of plane waves in the y-z (range-depth)
plane is assumed, in a time-invariant frozen ocean with D0 as the
water depth. All the angles are measured in the counterclockwise
direction with respect to the positive direction of "y". We model the
rough sea bottom and its surface as collections of Nb and Ns
scatterers, respectively, such that Nb>>1 and
Ns>>1. The i-th bottom scatterer is represented by
Sib, i=1, 2, . . . , Nb, whereas Sms denotes the
m-th surface scatterer, m=1, 2, . . . , Ns. Rays scattered from the
bottom and the surface are shown by solid thick and solid thin lines,
respectively. The rays scattered from Sib hit Rx1 and
Rx2 at the angle-of-arrivals (AOAs) γi,1b and
γi,2b, respectively. The traveled distances are labeled
by ξi,1b and ξi,2b, respectively. Similarly,
the scattered rays from Sms impinge on Rx1 and Rx2 at
the AOAs γm,1s and γm,2s, respectively,
with ξm,1s and ξm,2s as the traveled distances
shown in FIG. 3.

[0166]Let τ and γ represent the delay (travel time) and the AOA
(measured with respect to the positive direction of y, counterclockwise).
Then in the angle-delay domain, the impulse responses of the pressure
subchannels Tx-Rx1 and Tx-Rx2, represented by
χ1(γ,τ) and χ2(γ,τ), respectively,
can be written as:

[0167]In eqs. (7) and (8), δ(.) is the Dirac delta,
aib>0 and ams>0 represent the amplitudes of the
rays scattered from Sib and Sms, respectively,
whereas ψibε[0, 2π) and
ψmsε[0, 2 g) stand for the associated phases. The
four delay symbols in (7) and (8) represent the travel times from the
bottom and surface scatterers to the two receivers. For example,
τi,1b denotes the travel time from Sib to
Rx1, and so on. As becomes clear in Appendix I, the factors
(Nb)-1/2 and (Ns)-1/2 are included in (7) and (8),
and the subsequent channel functions, for power normalization. Also
0≦Λb≦1 represents the amount of the
contribution of the bottom scatterers. A close-to-one value for
Λb indicates that most of the received rays are being
received from the sea bottom. In this situation, the contribution of the
surface scatterers is given by 1-Λb.

[0168]A Dirac delta in the angle domain such as δ(γ-{tilde
over (γ)}) corresponds to a plane wave with the AOA of {tilde over
(γ)}, whose equation at an arbitrary point (y,z) ({tilde over
(γ)},{tilde over (z)}) is exp(jk[{tilde over (γ)} cos({tilde
over (γ)})+{tilde over (z)} sin({tilde over (γ)})]). For
example, δ(γ-γi,1b) in Eq. (7) represents
exp(jk[γ cos(γi,1b)+z
sin(γi,1b)])|y=0,z=z1=exp(jk z1
sin(γi,1b)). This is a plane wave emitted from the
scatter Sib that impinges Rx1, located at y=0 and
z=z1, through the AOA of γi,1b. Using similar plane
wave equations for the other angular delta functions in Equations (7) and
(8), the impulse responses of the pressure sub-channels Tx-Rx1 and
Tx-Rx2 in the delay-space domain can be respectively written as:

[0169]The terms y and z in equations (9) and (10) are intentionally
preserved, as in the sequel we need to calculate the spatial gradients of
the pressure, to obtain the velocities. Note that based on the definition
of the spatial Fourier transform, p1(τ) and p2(τ) can
be considered as the spatial Fourier transforms of
χ1(γ,τ) and χ2(γ,τ), respectively,
with respect to γ. Here we have assumed a constant sound speed
throughout the channel, which in turns make k fixed as well.

[0170]By taking the Fourier transform of equations (9) and (10) with
respect to τ, we respectively obtain the transfer functions of the
pressure subchannels Tx-Rx1 and Tx-Rx2 in the frequency-space
domain

[0172]Following the definition of the pressure-equivalent velocity in (2),
the velocity channels of interest in the delay-space and frequency-space
domains can be written as

ply(τ)=(jk)-1{dot over (p)}l(τ),
plz(τ)=(jk)-1p'l(τ), l=1,2, (13)

Ply(f)=(jk)-1{dot over (P)}l(f),
Plz(f)=(jk)-1P'l(f), l=1,2, (14)

[0173]where p1(τ) and P1(f), l=1,2, are given in Equations
(9)-(12). Furthermore, dot and prime denote the partial spatial
derivatives ∂/∂y and
∂/∂z, respectively. Clearly for l=1,2,
p1y(τ) and p1z(τ) are the pressure-equivalent
impulse responses of the velocity subchannels in the y and z directions,
respectively. Furthermore, P1y(f) and P1z(f)
represent the pressure-equivalent transfer functions of the velocity
subchannels in they and z directions, respectively, with l=1, 2.

A General Framework for Calculating the Correlations in Multipath Shallow
Waters

[0174]In a given shallow water channel, the numerical values of all the
amplitudes, phases, AOAs and delays in equations (9)-(12) are complicated
functions of many environmental characteristics such as the irregular
shape of the sea bottom and its layers/losses, volume microstructures,
etc. Due to the uncertainty/complexity in exact determination of all
these variables, we model them herein as random variables. More
specifically, we assume all the amplitudes {aib}i and
{ams}m are positive uncorrelated random variables,
uncorrelated with the phases {ψib}, and
{ψms}m. In addition, all the phases
{ψib}i and {ψms}m are uncorrelated,
and uniformly distributed over (0, 2π). The statistical properties of
the AOAs and delays will be discussed later. Overall, all the pressure
and velocity channel functions in equations (9)-(14) are random processes
in space, frequency and delay domains. In the following, we first derive
a closed-form expression for the pressure frequency-space correlation.
Then we show how other correlations of interest, which determine the
performance of a vector sensor receive array, can be calculated from the
pressure frequency-space correlation.

[0175]The Pressure Frequency-Space Correlation: We define this correlation
as Cp(Δf, L)=E[P2(f+Δf)P1*(f)]. The
correlation is expressed as follows:

[0176]This is a general pressure frequency-space correlation model that
holds for any AOA PDFs (Probability Density Functions) that may be chosen
for wbottom(γb) and wsurface(γs) In what
follows, first we use equation (15) to derive expressions for some
important spatial and frequency correlations, which hold for any AOA PDF.
These formulas provide useful intuition under rather general conditions.
Thereafter, we use a flexible parametric angular PDF for the AOA, to
obtain easy-to-use and closed-form expressions for correlations of
practical interest.

Spatial Correlations

[0177]The Pressure Correlation: At a fixed frequency with Δf=0, the
spatial pressure correlation can be obtained from equation (15) as:

[0178]where the overall AOA PDF w(γ) is defined as follows, to
include both the bottom and surface AOAs

w(γ)=Λbwbottom(γ)+(1-Λb)wsur-
face(γ). (17)

Of course, wbottom(γ)=0 for π<γ<2π, whereas
wsurface(γ)=0 for 0<γ<π. We keep equation (16)
as it is, without replacing εy by zero, since, later on, we
will take the derivative of Cp(0,L) with respect to εy
first.

[0179]The Pressure-Velocity Correlations: First we look at the z-component
of the velocity. Here we are interested in
E[P2(f){P1z(f)}*]=(-jk)-1
E[P2(f){P1'(f)}*], where P1z(f) is replaced according
to equation (14). On the other hand, similar to equation (5) one has
E[P2(f){P1'(f)}*]=-∂E[P2(f)P1*(f)]/.diff-
erential.L=-∂CP, (0,L)/∂L. Therefore:

[0180]where the integral in equation (18) comes from equation (16). An
interesting observation can be made when w(γ) is even-symmetric
with respect to γ=π or (symmetry of the AOAs from the bottom and
the surface with respect to the horizontal axis y). Then with L=0 in
equation (18) we obtain E[P1(f){P1z(f)}*]=0, i.e., the
co-located pressure and thez-component of the velocity are uncorrelated.

[0181]Now we focus on the y-component of the velocity. The correlation of
interest is
E[P2(f){P1y(f)}*]=(-jk)-1E[P2(f){{dot over
(P)}1(f)}*], where P1y(f) is replaced according to
equation (14). Note that according to the representations for P2(f)
and P1(f) in equations (27) and (28), respectively, the location of
the second sensor can be thought of as (y,z)=(εy,
z1+L), as εy→0, whereas the first sensor is
located at (y,z)=(0,z1). So, by considering the analogy of equation
(5) in they direction we obtain E[P2(f){{dot over
(P)}1(f)}]=-∂E[P2(f)P1*(f)]/∂.ep-
silon.y as εy→0=-∂CP(0,
L)/∂εy as εy→0.
Differentiation of equation (16) with respect to εy results
in

[0182]If w(γ) is even-symmetric around γ=π/2 and also
γ=3π/2, then with L=0 in equation (19) we obtain
E[P1(f){P1y(f)}*]=0, i.e., the co-located pressure and the
y-component of the velocity become uncorrelated.

[0183]The Velocity Correlations: Here we start with the z-component of the
velocity. We are going to calculate
E[P2z(f){P1z(f)}*]=k-2E[P2'(f){P1'(f)}-
*], where P2z(f) and P1z(f) are replaced according to
equation (14). On the other hand, similar to (6) one can write
E[P2'(f){P1'(f)}*]=-∂2E[P2(f)P1*(f)-
]/∂L2=-∂2C1'(0,
L)/∂L2. Hence

[0184]where equation (16) is used to write the integral in equation (20).

[0185]Attention is now directed to the y-component of the velocity. In
this case, the correlation is
E[P2y(f){P1y(f)}*]=k-2E[{dot over
(P)}2(f){{dot over (P)}1(f)}*], in which P2y(f) and
P1y(f) are replaced using equation (14). As discussed above,
the second and the first sensors are located at (y,z)=(εy,
z1+L), as εy→0, and (y,z)=(0,z1),
respectively. Thus, by using the equivalent of equation (6) in they
direction, we get E[{dot over (P)}2(f){{dot over
(P)}1(f)}*]=-∂2E[P2(f)P1*(f)]/.different-
ial.εy2 as
εy→0=-∂2CP(0,
L)/∂εy2 as εy→0. By
taking the second derivative of equation (16), with respect to
εy we obtain

[0186]The (average) received powers via the pressure-equivalent velocity
signals in the z and y directions are E[|P1z(f)|2] and
E[|P1y(f)|2], respectively. Using equations (20) and (21)
with L=0, and since sin2(γ)<1 and cos2(γ)<1,
one can easily show

E[|P1z(f)|2]<1, E[|P1y(f)|2]<1,
E[|P1z(f)|2]+E[|P1y(f)|2]=1. (22)

[0187]Therefore, the received powers via the two velocity channels are not
equal. However, via both of them together we receive the same total power
that a pressure sensor collects, as shown in equation (22). Note that in
this paper, the power received by a pressure sensor is
E[|P1(f)|2]=CP(0,0)=1, obtained from equation (16).

Closed-Form Correlations Using the Von Mises PDF

[0188]Here we propose to use two Von Mises PDFs for the bottom and surface
AOAs, as shown below

[0189]Each Von Mises PDF has two parameters: κb and μb
that control the angle spread and the mean AOA from the bottom,
respectively, whereas κs and μs represent the angle
spread and the mean AOA from the surface, respectively. In eq. (23),
I0 stands for the zero-order modified Bessel function of the first
kind. The von Mises PDF has proven to be useful in modeling the AOA and
calculating a variety of correlation functions in wireless multipath
channels. By substituting equation (23) into equation (16), and using the
following integral,

∫ π π exp ( αsinφ + βcosφ )
φ = 2 π I 0 ( α 2 +
β 2 ) , ( 24 )

[0190]the integral in equation (16) can be easily solved, which results in

[0191]According to equation (25), it is easy to verify that
CP(0,0)=1, consistent with the simplifying convention of unit (total
average) received pressure power. By taking the derivatives of equation
(25) with respect to L and εy, the closed-form expression
for a variety of correlations in vector sensor receivers can be obtained.

[0192]Below, equations (26) and (27), referred to above, are provided:

[0193]In general, there are two types of vector sensors: inertial and
gradient. Inertial sensors truly measure the velocity by responding to
the acoustic particle motion, whereas gradient sensors employ a
finite-difference approximation to estimate the gradients of the acoustic
field such as the velocity. Each sensor type has its own advantages and
disadvantages. Depending on the application, system cost, and required
precision, one can choose the proper sensor type.

[0194]In this section we derive basic system equations for data detection
via a vector sensor. To demonstrate the basic concepts of how both the
vector and scalar components of the acoustic field can be utilized for
data reception, we consider a simple system in a two-dimensional (2D)
depth-range underwater channel. As shown in FIG. 4, there is one transmit
pressure sensor, Tx, shown by a black dot, whereas for reception we use a
vector sensor, Rx, shown by a black square, which measures the pressure
and the y and z components of the particle velocity. This is basically a
1×3 SIMO system. With more pressure transmitters, one can have a
multiple-input multiple-output (MIMO) system, the discussion of which is
not provided in this section.

Pressure and Velocity Channels and Noises

[0195]Three channels are used in FIG. 4: the pressure channel p,
represented by a straight dashed line, and two pressure-equivalent
velocity channels pz and py, shown by curved dashed lines. To
define pz and py, we need to define the two velocity channels
vz and vy, the vertical and horizontal components of the
particle velocity, respectively. According to the linearized momentum
equation, the z and y component of the velocity at the frequency f0
are given by

vz=-(jρ0ω0)-1∂p/∂z-
, vy=-(jρ0ω0)-1∂p/.differential-
.y. (28).

[0196]In the above equations, ρ0 is the density of the fluid,
j2=-1 and ω0=2πf0. Eq. (28) simply states that
the velocity in a certain direction is proportional to the spatial
pressure gradient in that direction. To simplify the notation, the
velocity channels in equation (28) are multiplied by -ρ0c, the
negative of the acoustic impedance of the fluid, where c is the speed of
sound. This gives the associated pressure-equivalent velocity channels as
pz=-ρ0cvz and py=-ρ0cvy. With
λ as the wavelength, and k=2π/λ=ω0/c as the
wave number we finally obtain

pz=(jk)-1∂p/∂z,
py=(jk)-1∂p/∂p. (29)

[0197]The additive ambient noise pressure at the receiver is shown by n in
FIG. 4. At the same location, the z and y components of the ambient noise
velocity, sensed by the vector sensor are
ηz=-(jρ0ω0)-1∂n/.different-
ial.z and ηy=-(jρ0ω0)-1∂n/.-
differential.y, respectively, derived as shown in equation (28). So, the
vertical and horizontal pressure-equivalent ambient noise velocities are
nz=-ρ0cηz=(jk)-1∂n/.differential-
.z and ny=-ρ0cηy=(jk)-1∂n/.differ-
ential.y, respectively, which resemble equation (29).

Input-Output System Equations

[0198]According to FIG. 4, the received pressure signal at Rx in response
to the signal s transmitted from Tx can be written as r=p⊕s+n, where
⊕ stands for convolution in time. We also define the z and y
components of the pressure-equivalent received velocity signals as
rz=(jk)-1∂r/∂z and
ry=(jk)-1∂r/∂y, respectively. Based
on equation (29), and by taking the spatial gradient of r with respect to
z and y, we easily obtain the key system equations:

r=p⊕s+n, ry=py⊕s+ny, rz=pz⊕s+nz.
(30)

[0199]It is noteworthy that the three output signals r, ry and
rz are measured at a single point in space.

Pressure and Velocity Noise Correlations

[0200]We define the spatial pressure noise correlation between the two
locations (y+ly,z+lz) and (y,z) as qn(ly,
lz)=E[n(y+ly, z+lz)n*(y,z)], where the "*" operator
corresponds to a complex conjugate operation, and where ly and
lz are real numbers. Using the correlation properties of a
differentiator at the location (y,z) one can show
E[n{ny}*]=(jk)-1∂qn/∂ly,
E[n{nz}]=(jk)-1∂qn/∂lz and
E[nz{ny}*]=-k-2∂2qn/∂l-
z∂ly, all calculated for (ly, lz)=(0,0).
For an isotropic noise field in the y-z plane, we have qn(ly,
lz)=J0(k(ly2, +lz2)1/2), with
Jm(.) as the m-order Bessel function of the first kind. Using the
properties of the Bessel functions and their derivatives, it is easy to
verify that E[n{ny}*]=E[n{nz}*]=E[nz{ny}*]=0; i.e.,
all the noise terms in equation (30) are uncorrelated.

[0201]Below, the above noise correlations are derived, to demonstrate
under what conditions the noise terms in equation (30) are uncorrelated.

Pressure and Velocity Average Powers

[0202]Noise Powers: Using the statistical properties of a differentiator,
the powers of the y and z components of the pressure-equivalent noise
velocity at (y,z) can be obtained as
Ωny=E[|ny|2]=-k-2∂2q.sub-
.n/∂ly2 and
Ωnz=E[|nz|2]=-k-2∂2q.sub-
.n/∂Lz2, respectively, both calculated at (ly,
lz)=(0,0). Based on the qn of the 2D isotropic noise model
described previously, one can show that
Ωny=Ωnz=1/2. Note that the noise pressure
power in this model is Ωn=E[|n|2]=qn(0,0)=1. This
means that Ωn=Ωny+Ωnz.

[0203]Channel Powers: The ambient noise is a superposition of several
components coming from different angle of arrivals (AOAs). In multipath
environments such as shallow water, the channel is also a superposition
of multiple subchannels. Based on this analogy between n and p, as well
as their spatial gradients, one can obtain
Ωp=Ωpy+Ωpz, where
Ωp=E[|p|2], Ωpy=E[|py|2] and
Ωpz=E[|pz|2]. Note that in the 2D isotropic noise
model the distribution of AOA is uniform over the range (0, 2π), which
results in Ωny=Ωnz=Ωn/2.
However, this is not necessarily the case in multipath channels such
shallow waters, which means Ωpy and Ωpz
are not equal in general.

Multichannel Equalization with a Vector Sensor

[0204]In this section we use the basic zero forcing equalizer, to
demonstrate the feasibility of multichannel equalization with a compact
vector sensor receiver. Of course there are different types of equalizers
and we are not suggesting the zero forcing algorithm is the only possible
equalization method. However, since here the emphasis is not on equalizer
design, we have just used a simple equalizer to verify the concept. Thus,
this approach demonstrates the feasibility of multichannel
inter-symbol-interference (ISI) removal with a compact vector sensor
receiver. The system equation is

[0205]In equation (31), S=[s0 . . . sk-1]T includes K
transmitted symbols, and the symbol T refers to the transpose
operation. With M as the number of channel taps, the same for all 1,
l=1,2,3, R1=[r1(0) . . . r1(K+M-2)]T and
N1=[n1(0) . . . n1(K+M-2)]T are the 1-th
(K+M-1)×1 received signal and noise vectors, respectively. Also the
1-th (K+M-1)×K banded channel matrix is:

H l = [ h l ( 0 ) h l ( 0 )
h l ( M - 1 ) h l ( M - 1 )
] . ( 32 )

[0206]Note that for a vector sensor receiver, the channel indices 1, 2 and
3 in equation (31) represent the pressure, pressure-equivalent horizontal
velocity and pressure-equivalent vertical velocity, respectively. So,
based on equation (30), for an arbitrary discrete time index t, we have
r1(t)=r(t), r2(t)=ry(t), r3(t)==rz(t),
h1(t)=p(t), h2(t)=py(t), h3(t)=pz(t),
n1(t)=n(t), n2(t)=ny(t) and n3(t)=nz(t).
Assuming perfect channel knowledge at the receiver, the zero forcing
equalizer is

=(H.sup.†H)-1H.sup.†R, (33)

[0207]with S as the estimate of S and .sup.† as the transpose
conjugate (when H is not known at the receiver, one can use many
different methods to estimate H). The simulations of the following
section show the performance of equation (33).

Simulation Set up and Performance Comparison

[0208]Here we compare the performance of the vector sensor equalizer in
equation (33) with a vertical three-element pressure-only uniform linear
array (ULA) that performs the zero forcing equalization. The ULA
equations and equalizer are the same as in equations (31) and (33),
respectively, where the three channels represent three vertically
separated pressure channels. The noise vectors N1, N2 and
N3 in both receivers are considered to be complex Gaussians with
white temporal auto- and cross-correlations. For the isotropic noise
model discussed in subsection II-C, the noise vectors N1, N2
and N3 are uncorrelated in the vector sensor receiver. For the
pressure-only ULA with the element spacing of λ, there are some
small pressure correlations of J0(kλ)=0.22 and
J0(2kλ)=0.15 for the separations of λ and 2λ,
respectively, that are not included in the simulations. To calculate the
velocity channel impulse responses (IRs) py and pz in
simulations using the p channel IR generated by Bellhop, each spatial
gradient in equation (29) is approximated by a finite difference.
Therefore at location (y,z) we have
∂p(y,z)/∂z≈[p(y,
z+0.2λ)-p(y,z)]/(0.2λ) and
∂p(y,z)/∂y≈[p(y+0.2λ,z)-p(y,z)]/(-
0.2λ). Certainly one may devise other methods, to estimate the
velocity channel impulse responses. Here a simple technique was employed
to demonstrate the concept.

[0209]With an S vector that includes K=200 equi-probable ±1 symbols,
and the noise vector and channel matrix N and H generated as described
above, the received vector R is calculated using equation (31). Then S is
estimated using equation (33), and the bit error rate is shown in FIG. 6.
(FIG. 5 shows the bit error rate for various sensor types in a different
embodiment, in a frequency-flat channel). The water depth for the shallow
channel of FIG. 6 is 81.1 meters (m), where the Tx and Rx are 5
kilometers (km) apart. For this simulation, The Tx and Rx are 25 m and 63
m below the water surface, respectively. A coarse silt bottom is
considered, with f0=12 kHz and a bit rate of 2400 bits/sec. More
detailed information about the channel, the measured sound speed profile
and the results of other channels are presented later in this document.

[0210]To define the average signal-to-noise ratio (SNR) per channel in
FIG. 6, let p=[p(0) . . . p(M-1)]T, py=[py(0) . . .
py(M-1)]T and pz=[pz(0) . . . pz(M-1)]T be
the taps of the pressure, y- and z-velocity IRs, respectively. Then the
pressure, y- and z-velocity SNRs are
ζp=Ωp/Ωn,
ζpy=Ωpy/Ωny and
ζpz=Ωpz/Ωnz, respectively,
such that Ωp=p.sup.†p,
Ωpy=(py).sup.†py and
Ωpz=(pz).sup.†pz. The average SNR per
channel for the vector sensor receiver is
ζ=(ζp+ζpy+ζpz)/3 by
definition. Also p is normalized such that Ωp=1. This implies
that Ωpy+Ωpz=1 in our simulations. Since
Ωny=Ωnz=Ωn/2 in a 2D isotropic
noise model, we finally obtain ζ=1/Ωn, which is the same
as the SNR of a unit-power pressure channel ζp.

Discussion of Results

[0211]The performance of an embodiment of the compact vector sensor
receiver in comparison with prior art sensors is shown in FIG. 6. It may
be readily observed that the bit error rate (shown using a logarithmic
scale on the vertical axis) of the vector sensor is significantly lower
than the bit error rates of the single pressure sensor and of the
pressure sensor arrays over the entire range of signal to noise ratio
(SNR) per channel (as shown along the horizontal axis). Moreover, at
higher SNR levels, such as at 6-7 dB, the vector sensor bit-error rate is
about an order of magnitude lower than those of the various pressure
sensor approaches. Therefore, a considerable performance improvement is
obtained employing the vector sensor system and method disclosed herein.

[0212]Performance of two three-element pressure-only arrays with element
spacings of λ and 2λ are also shown, which in this
simulation, are slightly worse than a vector sensor receiver. By changing
the simulation scenario, for example the sea-bottom type, one may observe
a better performance for the pressure-only array. However, even in such
cases, both the vector sensor and pressure-only array receivers exhibit
better performance than a single pressure sensor receiver. Moreover, the
vector sensor exhibits better performance than the pressure-only array
receiver embodiments (whether using λ or 2λ spacing) and
offers the further benefit of providing this superior bit error rate
performance within a smaller physical package.

DETAILED SIMULATION STUDY

Introduction to Simulation Discussion

[0213]The underwater communications channel is characterized as a
multipath channel. The ensemble average channel impulse responses of the
underwater communications channels were determined using Acoustic
Toolbox.

[0214]The first part of the project modified the acoustic toolbox to be
able to plot an impulse response of an underwater channel at a given
location and to save the numerical details of the impulse responses from
the resulting arrival file. Next, using the information of the channel
impulse responses from the first part, Monte Carlo Simulations were
performed which transmitted Binary Phase Shift Keying (herein referred to
as BPSK) signals through underwater communication channels. The signals
were the received using an SISO ZF (zero forcing) receiver. An
improvement in the performance of the system was observed when SIMO ZF
receivers were used. Further, alternative ways to receive the signal are
outlined, including using one pressure sensor and one vector sensor to
detect the signals.

Acoustic Toolbox Modifications

Introduction to ACT

[0215]Matlab Acoustic Toolbox, or "ACT", was used for the simulations
described herein. ACT is a menu-based user interface for running a number
of underwater acoustic propagation models and plotting the results. An
initial deficiency of ACT was that information about channel impulse
response at given locations could not be directly obtained. Channel
impulse response information was found by using the Bellhop underwater
acoustic propagation model on amplitude-delay mode. The resulting arrival
file (*.arr) was used to plot an impulse response at a given location and
to save the numerical data of the channel impulse responses of all
locations in the arrival file ("arr" file). From the "arr" file, the
information on the impulse response provided the ensemble average impulse
response.

Geographic Representation in ACT

[0216]Throughout this report, the geographic representations as shown in
FIG. 7 are referred to. FIG. 8 below shows the menus of the ACT toolbox.
An additional menu tap (button or bar) for "Impulse Response" has been
added in the "Plotting Options" menu. Once the Impulse Response tap was
pressed, another submenu popped up, providing three options which were 1)
Plot the impulse Response, 2) Save the Original Impulse Response to file,
and 3) Back to the Plotting Menu.

Plotting the Impulse Response

[0217]If option I (of the above three options) is chosen, an "arr" file
needed to be selected, and then the geographical details of a transmitter
and a receiver needed to be specified on the menus in FIG. 9.

[0218]After inputting all the details, the figure of the channel impulse
response at the particular location was displayed, along with the choice
to save the detail information of the impulse response into a
text-formatted file having a ".mat" extension, as shown in FIG. 10. A
root name for the file was entered to save the information associated
with each impulse response. The information below is a shortened version
of the information contained in the resulting saved file with impulse
response information. In the following, some of the multipath component
information has been removed.

[0219]The first, second, and third columns below represent the delay time,
the magnitude of the impulse response and its phase delays in radians,
respectively:

[0220]Applying option 2 described above, the information of impulse
responses in the selected arr file was separately saved in text-formatted
files with mat extension according to geographic locations. First, the
prefix root name of the save file was specified.

[0221]The nomenclature of the saved file was based on the information of
the prefix root name, source depth, and receiver depth. For example, if
the chosen "arr" file contained the impulse response information at a
source depth equal to 25 m and receiver depths equal to 60 m and 63 m,
and the receiver ranges were 7000 m and 7001 m, the saved files had the
following file names given that the prefix root name was xxxx:

xxxx_IRatSD25RD60.mat and xxxx_IRatSD25RD63.mat

[0222]The information below is an example of the structure of each
resulting saved file.

[0223]These data are represented in tri-column groups for the receiver
range(s): 7000 7001 m, respectively. Each group has three columns where
the first corresponds for the delay in seconds, the second for amplitude,
and the third for phases in radians.

[0224]In some cases there might be zeros at the bottom of the matrix.
These zeros are excluded to the IR data.

[0225]The first three columns, in each row above, contain information for
the impulse response at a receiver range of 7000 m while the second three
columns or columns 4, 5, and 6 provide the information on the impulse
response at the receiver range 7001 m.

[0226]In other words, the data on the impulse responses were divided into
tri-column groups and the number of groups is equal to the number of the
receiver ranges. Each group has three columns where the first corresponds
for the delay in seconds, the second for amplitude, and the third for
phases in radians.

Reading the Information from the Resulting Saved File

[0227]The command shown below is an example of a command to read all the
numerical detail of the impulse response into a matrix in Matlab.

M=textread(`xxxx.mat`,` `,`commentstyle`, `matlab`);

Performance of Multichannel Underwater Communications Receivers

[0228]Herein, the performance of transmitting Binary Phase Shift Keying is
determined using a Monte Carlo Simulation and a BPSK signal with
rectangular pulse shape through underwater communications channels.
Acoustic waves were used for this underwater application, unlike in air
channels where electromagnetic waves would generally be used. Using a
transducer, the transmitter converted the electrical signals into
pressure signals and the receiver reconverted the pressure signals back
into electrical waveforms.

[0229]Assuming knowledge of the channels at the receivers, a SISO ZF
receiver and SIMO ZF receivers were used. When using the SIMO ZF
receivers, a combination of sensors was used which included: two pressure
sensors, (P-P) separated with the distances of 0.2λ, λ, and
2λ, and a pressure sensor with a velocity vector sensor, (P-V),
separated with the distance of 0.2λ in both the horizontal and
vertical directions.

[0230]Assumed system parameters included the central frequency of the
carrier fc=12 kHz, the sampling frequency of the channel fs=48
kHz, and the data rate Rb=2,400 kbps.

Underwater Communications Channel

[0231]An Underwater Communications Channel can be categorized as a
multipath channel. Its impulse response has a time-varying property as
depicted in FIG. 11. In this paper, the channel impulse response was
simulated using ACT toolbox resulting in the ensemble average of the
channel.

Time Dispersion Parameters

[0232]To compare the differences among multipath channels and to develop
some design concepts for wireless systems, parameters that quantify the
multipath channel were used. The mean excess delay, rms delay spread, and
maximum excess delay spread are multipath channel parameters that can be
determined from a power delay profile. The mean excess delay was given by

[0233]The maximum excess delay (XdB) was defined to be the time delay
during which multipath energy fell to X dB below the maximum value.

The τ-spaced model

[0234]For computer simulation purposes, it's useful to discretize the
multipath delay axis r into N excess delay bins which have equal delay
segments, Δτ, and to shift the first arrival multipath
component into τ0. Depending on the choice of Δτ and
physical channel delay properties, there may be two or more multipath
components that fall into the same excess delay bin that have to be
vectorially combined to yield the instantaneous response. The discretized
impulse response is called a τ-spaced impulse response.

The T-Spaced model

[0235]Usually, the baud duration T of a typical digital communication
system is longer than delay segment Δτ of τ-spaced channel.
We can dramatically reduce the simulation time by setting the simulation
step size to the baud duration T. FIG. 12 depicts the method for
generating correlated T-spaced tap coefficients from a τ-spaced
impulse response.

Velocity Vector Sensor

[0236]The velocity of the acoustic wave is defined as the spatial gradient
of its pressure. In the following, a method for using velocity vector
sensor for data demodulation and equalization is employed. Assuming that
h1, h2, and h3 are the pressure impulse responses of the
particular locations in FIG. 13, where h1 is the pressure impulse
response at the receiver location, the velocity impulse responses are
given by:

in the horizontal direction,

v y = h 1 - h 3 dy ( 37 )

and in vertical direction

v z = h 1 - h 2 dz ( 38 )

It is noted that in this paper dy=dz=0.22λ

SISO and SIMO Signal Models

SISO: Single Input Single Output

[0237]The input-output relation of a SISO frequency selective channel is
represented as follows:

Y [ k ] = E s HS [ k ] + N [ k ] ( 39
)

where Es is defined as the energy per symbol.

[0238]Preferably, the impulse response vector h equals 1×L and is
expressed as

h=[h[0] . . . h[L-1]] (40)

[0239]To ensure that the channel does not artificially amplify or
attenuate the signal, h is normalized so that Σ|h|2=1[6]. H is
(T+L-1)×T where T is defined as the number of symbols and is
written as:

H = [ h [ 0 ] 0 0 h [ L - 1
] 0 0 h [ 0 ] 0
0 h [ L - 1 ] ] ( 41 )

[0240]The vectors Y[k] and N[k] are (T+L-1)×1 where T equals the
number of symbols, while S[k] is T×1. N[k] is a complex-valued
Gaussian noise with zero mean and N0 variance. It should be noted
that T=200 in this paper.

SIMO: Single Input Multiple Outputs

[0241]Using Eq. (39), for a SISO channel, the SIMO channel was written as

[0244]The vectors Y[k] and N[k] had dimension MR(T+L-1)×1,
whereas H was MR(T+L-1)×T. It's also noted that for the SIMO
in this paper MR=2.

Zero Forcing Receiver

[0245]The goal of using the ZF receiver was to invert the channel and
eliminate ISI. First, the signal was received using a matched filter and
then equalized. The output of the ZF equalizer was given by:

R[k]=(HH H)-1HHY[k] (45)

where vector R[k] was T×1.

Signal Detection

[0246]An example of the signal constellation of R[k], the output signal of
the ZF equalizer when BPSK signal was transmitted, is shown in FIG. 14.
Throughout this paper, the BPSK signals were assumed to be equally
probable, so the decision rule for the optimum detector was as follows:
Deciding that bit 1 was sent if Re(R[k])>0 and that bit 0 was sent if
Re(R[k])<0, where Re(. .) refers to real part of the complex number.

Performance Analysis of ZF receiver

[0247]From Equation (45), the SNR at the output of the equalizer was given
by

S N R k = ( Eb / N 0 ) k [ H H H ]
k , k - 1 , for k = 1 , , T ( 46 )

Channel Simulation Parameters and Notations

Geographic Representations

[0248]Throughout the simulations, it was assumed that the transmitter was
located at 25 m below the water level while the locations of the
receivers varied within a 9-by-9 meter-squared region at the initial
receiver ranges of 5 km and 10 km.

[0249]According to FIG. 15, there are 16 receiver locations at a
particular receiver range. The receiver locations were assigned a number
in the same way as the elements of a 4-by-4 matrix, for example, for a
receiver location rij, i=1, . . . , 4, j=1, . . . , 4. Thus, thus
receiver locations in FIG. 15 include rows 1 through 4, with each such
row having four locations numbered 1 through 4. Herein, each receiver
location is referred to using a two-digit format, with the first digit
designating the row, and the second digit designating the "column" of
that receiver location in the two-dimensional array shown in FIG. 15.
Thus, receiver location "44", for instance, refers to the receiver
location at the lower right extreme of the 4×4 receiver-location
array.

Sound Speed Profile

[0250]The sound speed profile in the simulation represented real channel
observations obtained on May 10, 2002 in waters off San Diego, Calif. The
properties of the water are shown in the Table 1 and the sound speed
profile is shown in Table 2.

[0251]It should be noted that sound speeds vary with depth, thus the
wavelength λ was also variable. Using SIMO equalization, the
wavelength values were used to determine the distance between receiver
antennas. An approximate λ was calculated assuming a constant sound
speed of 1,500 m/s, since the sound speeds vary slightly about that
value. This variation can be negligible when the center frequency is very
large. Therefore, λ was defined by:

λ = v f c = 1500 m / s 12000 Hz =
0.125 m . ( 47 )

Bottom Profile

[0252]Channel impulse responses were simulated using two bottom profiles,
coarse silt and very fine sand. Their properties of these bottom profiles
are shown in the Tables 4.3 and 4.4, respectively.

[0254]The impulse response under various conditions are shown in FIGS.
17-22 of this application, which FIGS. are described in the Brief
Description of the Drawings section of this specification. For the sake
of brevity, the drawing descriptions provided in the "Brief Description"
are not repeated in this section. This practice will be applied to
subsequent sections of this application for which the data is fully
provided in the drawings and in the Brief Description of the Drawings.

Mean Excess Delay and RMS Delay Spread

[0255]Below, data for mean excess delays in seconds of 16 receiver
locations are tabulated. In the following tables, the pertinent values
for the 16 receiver locations are arranged in the tables in accordance
with their distribution in the array shown in FIG. 15. Thus, receiver 11
data is located at the upper left of both the array and of the data
tables; and receiver 44 data is located at the lower right of both the
array and the data tables, and data for the other receiver locations are
distributed throughout the tables in positions corresponding to their
respective locations within the array of FIG. 15.

[0256]Table 5 tabulates the mean excess delays in seconds of the pressure
impulse responses of 16 receiver locations for the initial receiver range
5 km and coarse silt bottom profile, with the mean=1.2369e-002 sec and
the variance=5.1552e-006 sec2.

[0269]Bit error rate plots are shown in FIGS. 29-30, Eigen Value plots in
FIGS. 31-32, and Inverted Diagonal Elements of (HH H)-1 in
FIGS. 33-34 of this application, which FIGS. are described in the Brief
Description of the Drawings section of this specification.

The Condition Number of HH H

[0270]Table 17 tabulates the condition numbers of HH H of SISO ZF
receivers of 16 receiver locations for the initial receiver range 5 km
and coarse silt bottom profile, where the mean 350.3018, and the
variance=6.8536e+004.

[0271]Table 18 tabulates the condition numbers of HH H of SIMO ZF
receivers when using two pressure sensors separated at the distance of
0.2λ, of 16 receiver locations for the initial receiver range 5 km
and coarse silt bottom profile, where the mean=310.0675, and the
variance=2.5314e+004.

[0272]Table 19 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using two pressure sensors separated at the distance of
λ, for 16 receiver locations for the initial receiver range 5 km
and coarse silt bottom profile, where the mean=263.0749, and the
variance=1.3027e+004.

[0273]Table 20 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using two pressure sensors separated at the distance of
2λ, of 16 receiver locations for the initial receiver range 5 km
and coarse silt bottom profile, where the mean=221.5968, and the
variance=8.8299e+003.

[0274]Table 21 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using a pressure sensor and a horizontal velocity vector
sensor separated at the distance of 0.2λ, of 16 receiver locations
for the initial receiver range 5 km and coarse silt bottom profile, where
the mean=63.4770, and the variance=1.0247e+003.

[0275]Table 22 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using a pressure sensor and a vertical velocity vector
sensor separated at the distance of 0.2λ, of 16 receiver locations
for the initial receiver range 5 km and coarse silt bottom profile, where
the mean=113.7906, and the variance=6.3834e+003.

[0276]Graphs for the impulse response are shown in FIGS. 35-40 of this
application. Descriptions of FIGS. 35-40 are provided in the Brief
Description of the Drawings section of this specification. For the sake
of brevity, those descriptions are not repeated in this section.

Mean Excess Delay and RMS Delay Spread

[0277]Mean Excess Delays in Seconds of 16 Receiver Locations

[0278]Table 23 tabulates the mean excess delays in seconds of the pressure
impulse responses of 16 receiver locations for the initial receiver range
10 km and coarse silt bottom profile where the mean=1.2811e-002 sec; and
the variance=2.9050e-005 sec2.

[0285]Graphs of frequency responses are shown in FIGS. 41-46 of this
application. Descriptions of these figures are provided in the Brief
Description of the Drawing section of this application. For the sake of
brevity, those descriptions are not repeated in this section.

DC Average and Variance of Impulse Response

DC Average Over 16 Receiver Locations

[0286]Table 29 tabulates the DC average of pressure impulse responses of
16 receiver locations for the initial receiver range 10 km and coarse
silt bottom profile, where the mean=1.0296e-007-4.1454e-007i volts, and
the variance=2.4146e-014 volts2.

[0292]Bit error rate plots are provided in FIGS. 47-48 of this
application. Descriptions of these figures are provided in the Brief
Description of the drawings of this specification.

Eigen Values Plots

[0293]Eigen Values plots are provided in FIGS. 49-50 of this application.
Descriptions of these figures are provided in the Brief Description of
the drawings of this specification.

The Plots of Inverted Diagonal Elements of (HH H)-1

[0294]The plots of inverted diagonal elements of (HH H)-1 are
provided in FIGS. 51-52 of this application. Descriptions of these
figures are provided in the Brief Description of the drawings of this
specification.

The Condition Number of HH H

[0295]Table 35 tabulates the condition numbers of HH H of SISO ZF
receivers of 16 receiver locations for the initial receiver range 10 km
and coarse silt bottom profile, where the mean=138.8664, and the
variance=2.3184e+004.

[0296]Table 36 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using two pressure sensors separated at the distance of
0.2λ, of 16 receiver locations for the initial receiver range 10 km
and coarse silt bottom profile, where the mean=133.5162, and the
variance=2.0231e+004.

[0297]Table 37 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using two pressure sensors separated by a distance
λ, for 16 receiver locations for the initial receiver range 10 km
and coarse silt bottom profile, where the mean=130.3825, and the
variance=1.6308e+004.

[0298]Table 38 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using two pressure sensors separated at the distance of
2λ, of 16 receiver locations for the initial receiver range 10 km
and coarse silt bottom profile, where the mean=136.1120, and the
variance=1.7377e+004.

[0299]Table 39 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using a pressure sensor and a horizontal velocity vector
sensor separated at the distance of 0.2λ, for 16 receiver locations
for the initial receiver range 10 km and coarse silt bottom profile,
where the mean=24.4145, variance=233.0000.

[0300]Table 40 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using a pressure sensor and a vertical velocity vector
sensor separated at the distance of 0.2λ, of 16 receiver locations
for the initial receiver range 10 km and coarse silt bottom profile,
where the mean=25.7880, and the variance=180.2970.

[0301]Graphs of the impulse response under various conditions are shown in
FIGS. 53-58 of this application, and descriptions of these figures are
provided in the Brief Description of the Drawings section of this
specification.

Mean Excess Delay and RMS Delay Spread

[0302]Mean Excess Delays in Seconds of 16 Receiver Locations

[0303]Table 41 tabulates the mean excess delays in seconds of the pressure
impulse responses of 16 receiver locations for the initial receiver range
5 km and very fine sand bottom profile, where the mean=5.2933e-002 sec;
and the variance=1.7585e-005 sec2.

[0304]Table 42 tabulates the mean excess delays in seconds of the
horizontal velocity impulse responses of 16 receiver locations for the
initial receiver range 5 km and very fine sand bottom profile, where the
mean=7.3936e-002 sec; and the variance=2.2817e-003 sec2.

[0305]Table 43 tabulates the mean excess delays in seconds of the vertical
velocity impulse responses of 16 receiver locations for the initial
receiver range 5 km and very fine sand bottom profile, where the
mean=5.5926e-002 sec; and the variance=2.8826e-004 sec2.

[0307]Table 44 tabulates the RMS delays spreads in seconds of the pressure
impulse responses of 16 receiver locations for the initial receiver range
5 km and very fine sand bottom profile, where the mean=4.7823e-002 sec;
and the Variance=1.8346e-007 sec2.

[0308]Table 45 tabulates the RMS delays spreads in seconds of the
horizontal velocity impulse responses of 16 receiver locations for the
initial receiver range 5 km and very fine sand bottom profile, where the
mean=1.5964e-002 sec; and the variance=2.3930e-004 sec2.

[0309]Table 46 tabulates the RMS delays spreads in seconds of the vertical
velocity impulse responses of 16 receiver locations for the initial
receiver range 5 km and very fine sand bottom profile, where the
mean=4.5399e-002 sec; and the Variance=5.9946e-005 sec2.

[0310]Graphs of frequency response under various conditions are shown in
FIG. 59-64. Descriptions of these figures are provided in the Brief
Description of the Drawings section of this specification.

DC Average and Variance of Impulse Response

DC Average Over 16 Receiver Locations

[0311]Table 47 tabulates the DC average of pressure impulse responses of
16 receiver locations for the initial receiver range 5 km and very fine
sand bottom profile, where the mean=-8.5798e-008+4.4775e-008i volts, and
the variance=7.3867e-016 volts2.

[0312]Table 48 tabulates the DC average of horizontal velocity impulse
responses of 16 receiver locations for the initial receiver range 5 km
and very fine sand bottom profile, where the
mean=-1.0355e-009+1.2048e-009i volts, and the variance=2.2574e-017
volts2.

[0313]Table 49 tabulates the DC average of vertical velocity impulse
responses of 16 receiver locations for the initial receiver range 5 km
and very fine sand bottom profile, where the
mean=-4.9693e-009-1.1694e-008i volts, and the variance=1.1141e-015
volts2.

[0314]Table 50 tabulates the variance of pressure impulse responses of 16
receiver locations for the initial receiver range 5 km and very fine sand
bottom profile, where the mean=1.9048e-010 volts2, and the
variance=1.8119e-022 volts4.

[0315]Table 51 tabulates the variance of horizontal impulse responses of
16 receiver locations for the initial receiver range 5 km and very fine
sand bottom profile, where the mean=3.1669e-008 volts2, and the
variance=1.1129e-015 volts4.

[0317]Plots for bit error rate under various conditions are shown in FIGS.
65-66. Eigen values plots are shown in FIGS. 67-68. Plots for the
condition number of (HH H)-1 are shown in FIGS. 69-70. The
foregoing figures are described in the Brief Description of the Drawings
section of this specification.

5.3.8 The Condition Number of HH H

[0318]Table 53 tabulates the condition numbers of HH H of SISO ZF
receivers of 16 receiver locations for the initial receiver range 5 km
and very fine sand bottom profile, where the mean=73.5595, and the
variance=881.4456.

[0319]Table 54 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using two pressure sensors separated at the distance of
0.2λ, of 16 receiver locations for the initial receiver range 5 km
and very fine sand bottom profile, where the mean=71.9622, and the
variance=965.2747.

[0320]Table 55 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using two pressure sensors separated at the distance of
λ, of 16 receiver locations for the initial receiver range 5 km and
very fine sand bottom profile, where the mean=68.2852, and the
variance=927.5209.

[0321]Table 56 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using two pressure sensors separated at the distance of
2λ, for 16 receiver locations for the initial receiver range 5 km
and very fine sand bottom profile, where the mean=70.8836, and the
variance=1466.4923.

[0322]Table 57 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using a pressure sensor and a horizontal velocity vector
sensor separated at the distance of 0.2λ, for 16 receiver locations
for the initial receiver range 10 km and coarse silt bottom profile,
where the mean=48.7019, and the variance=359.8946.

[0323]Table 58 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using a pressure sensor and a vertical velocity vector
sensor separated at the distance of 0.2λ, for 16 receiver locations
for the initial receiver range 10 km and coarse silt bottom profile,
where the mean=58.04779, and the variance=499.6343.

[0324]Plots of the impulse response for the conditions stated the above
heading are shown in FIGS. 71-76 of this application. These figures are
described in the Brief Description of the Drawings section of this
specification.

Mean Excess Delay and RMS Delay Spread

Mean Excess Delays in Seconds of 16 Receiver Locations

[0325]Table 59 tabulates the mean excess delays in seconds of the pressure
impulse responses of 16 receiver locations for the initial receiver range
10 km and very fine sand bottom profile, where the mean=6.6085e-002 sec;
and the variance=6.7673e-004 sec2

[0326]Table 60 tabulates the mean excess delays in seconds of the
horizontal velocity impulse responses of 16 receiver locations for the
initial receiver range 10 km and very fine sand bottom profile, where the
mean=9.7303e-002 sec; and the variance=8.6938e-003 sec2.

[0327]Table 61 tabulates the mean excess delays in seconds of the vertical
velocity impulse responses of 16 receiver locations for the initial
receiver range 10 km and very fine sand bottom profile, where the
mean=9.4922e-002 sec; and the variance=1.9379e-003 sec2.

[0329]Table 62 tabulates the RMS delays spreads in seconds of the pressure
impulse responses of 16 receiver locations for the initial receiver range
10 km and very fine sand bottom profile, where the mean=8.2024e-002 sec;
and the variance=1.1088e-004 sec2.

[0330]Table 63 tabulates the RMS delays spreads in seconds of the
horizontal velocity impulse responses of 16 receiver locations for the
initial receiver range 10 km and very fine sand bottom profile, where the
mean=4.6591e-002 sec; and the variance=2.0241e-003 sec2.

[0331]Table 64 tabulates the RMS delays spreads in seconds of the vertical
velocity impulse responses of 16 receiver locations for the initial
receiver range 10 km and very fine sand bottom profile, where the
mean=8.4227e-002 sec; and the variance=3.7238e-004 sec2.

[0332]Graphs of the frequency response under various conditions are shown
in FIGS. 77-82. These figures are also described in the Brief Description
of the Drawings section of this specification.

DC Average and Variance of Impulse Response

[0333]DC Average over 16 Receiver Locations

[0334]Table 65 tabulates the DC average of pressure impulse responses of
16 receiver locations for the initial receiver range 10 km and very fine
sand bottom profile, where the mean=5.0168e-008+1.7987e-009i volts, and
the variance=1.2961e-015 volts2.

[0335]Table 66 tabulates the DC average of horizontal velocity impulse
responses of 16 receiver locations for the initial receiver range 10 km
and very fine sand bottom profile, where the
mean=4.3398e-009+1.6282e-008i volts, and the variance=4.1590e-015
volts2.

[0336]Table 67 tabulates the DC average of vertical velocity impulse
responses of 16 receiver locations for the initial receiver range 10 km
and very fine sand bottom profile, where the
mean=5.2747e-010+3.2734e-008i volts, and the variance=7.3608e-015
volts2.

[0337]Table 68 tabulates the variance of pressure impulse responses of 16
receiver locations for the initial receiver range 10 km and very fine
sand bottom profile, where the mean=8.8535e-011 volts2, and the
variance=1.9566e-021 volts4.

[0338]Table 69 tabulates the variance of horizontal impulse responses of
16 receiver locations for the initial receiver range 10 km and very fine
sand bottom profile, where the mean=3.4554e-009 volts2, and the
variance=1.6114e-017 volts4.

[0339]Table 70 tabulates the variance of vertical impulse responses of 16
receiver locations for the initial receiver range 10 km and very fine
sand bottom profile, where the mean=4.5194e-008 volts2, and the
variance=9.8903e-016 volts

[0340]Plots for bit error rate under various conditions are shown in FIGS.
83-84. Eigen values plots are shown in FIGS. 85-86. Plots for the
condition number of (HH H)-1 are shown in FIGS. 87-88. The
foregoing figures are described in the Brief Description of the Drawings
section of this specification.

5.4.8 The Condition Number of HH H

[0341]Table 71 tabulates the condition numbers of HH H of SISO ZF
receivers of 16 receiver locations for the initial receiver range 10 km
and very fine sand bottom profile, where the mean=23.2061, and the
variance=116.1863.

[0342]Table 72 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using two pressure sensors separated by a distance of
0.2λ, for 16 receiver locations for the initial receiver range 10
km and very fine sand bottom profile, where the mean=23.0478, and the
variance=110.5727.

[0343]Table tabulates the condition numbers of HH H of SIMO ZF
receivers when using two pressure sensors separated by a distance of
λ, for 16 receiver locations for the initial receiver range 10 km
and very fine sand bottom profile, where the mean=21.7560, and the
variance=80.7291.

[0344]Table 74 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using two pressure sensors separated by 2λ, for 16
receiver locations for the initial receiver range 10 km and very fine
sand bottom profile, where the mean=21.5075, and the variance=74.3992.

[0345]Table 75 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using a pressure sensor and a horizontal velocity vector
sensor separated by 0.2λ, for 16 receiver locations for the initial
receiver range 10 km and coarse silt bottom profile, where the
mean=13.0732, and the variance=34.0670.

[0346]Table 76 tabulates the condition numbers of HH H of SIMO ZF
receivers, when using a pressure sensor and a vertical velocity vector
sensor separated at the distance of 0.2λ, for 16 receiver locations
for the initial receiver range 10 km and coarse silt bottom profile,
where the mean=13.4015, and the variance=30.6241.

[0347]In this section, after examining the time dispersion properties of
the underwater channels, it has been determined that the rms delay
spreads vary within a range of about 1-10 milliseconds. This range
corresponds roughly to a range of about 20-200 Hz for 50% of the coherent
bandwidth of the channels. Since the signal needs to be transmitted at
the rate of 2,400 bps, there is a need for equalization.

[0348]In this experiment, the performance of P-P SIMO ZF receivers
exceeded that of the SISO ZF receiver. In general, the P-P SIMO ZF
receivers with two pressure sensors separated by the distance d=2λ
had a slightly better performance than those at d=0.2λ and λ.
Further, pressure sensors separated at distance d=λ performed
somewhat better than those separated by d=0.2λ.

[0349]The simulation results showed that using vector sensors in the SIMO
ZF receivers, separated from the pressure receiver by d=0.2λ,
improved the performance of the SIMO ZF receivers. More importantly, the
vector sensors reduced the volume of the receiver system. Further, the
simulation suggested that in general, P-Vy performs better than P-Vz.

[0350]It is noted that the methods and apparatus described thus far and/or
described later in this document may be achieved utilizing any of the
known technologies, such as standard digital circuitry, analog circuitry,
any of the known processors that are operable to execute software and/or
firmware programs, programmable digital devices or systems, programmable
array logic devices, DSP (digital signal processing) processors, or any
combination of the above. One or more embodiments of the invention may
also be embodied in a software program for storage in a suitable storage
medium and execution by a processing unit.

[0351]The applicant has attempted to disclose all embodiments and
applications of the disclosed subject matter that could be reasonably
foreseen. However, there may be unforeseeable, insubstantial
modifications that remain as equivalents. While the present invention has
been described in conjunction with specific, exemplary embodiments
thereof, it is evident that many alterations, modifications, and
variations will be apparent to those skilled in the art in light of the
foregoing description without departing from the spirit or scope of the
present disclosure. Accordingly, the present disclosure is intended to
embrace all such alterations, modifications, and variations of the above
detailed description. It is therefore to be understood that numerous
modifications may be made to the illustrative embodiments and that other
arrangements may be devised without departing from the spirit and scope
of the present invention as defined by the appended claims.